Probabalistic quantification of intermittently
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unstable dynamical systems
MAAHSETTSL INSTITUTE
OF TECHNOLOGY
by
OCT 01 2015
Mustafa A. Mohamad
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B.S., University of Illinois at Urbana-Champaign (2012)
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2015
@
Massachusetts Institute of Technology 2015. All rights reserved.
A
Author ..............
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Department of Mechanical Engineering
August 7215
Certified by............ ....
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is P. Sapsis
Th
anical Engineering
Assistant Professor of lc
Thesis Supervisor
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.
Accepted by
David E. Hardt
Professor of Mechanical Engineering
Chairman, Department Committee on Graduate Theses
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Probabalistic quantification of intermittently unstable
dynamical systems
by
Mustafa A. Mohamad
Submitted to the Department of Mechanical Engineering
on August 7, 2015, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
In this work we consider dynamical systems that are subjected to intermittent instabilities. The presence of intermittent instabilities can be identified by large amplitude
spikes in the time series of the system or heavy-tails in the probability density function
(pdf) of the response. We formulate a method that can analytically approximate the
response pdf (both the main probability mass and heavy-tail structure) for systems
where intermittency is important to quantify. The method relies on conditioning the
probability density function on the occurrence of an instability and the separate analysis of the two states of the system, the unstable state and the otherwise stable state
with no intermittent events, according to a total probability law argument.
In the stable regime we employ steady state assumptions, which lead to the derivation of the conditional response pdf using standard methods for random dynamical
systems. The unstable regime is inherently transient and to analyze this regime we
characterize the response under the assumption of an exponential growth phase and a
subsequent decay phase until the system is brought back to the stable attractor. The
separation into a stable regime and unstable regime, allows us to capture the heavytailed statistics associated with intermittent instabilities. We illustrate the method
on three prototype intermittent systems and show that the analytical approximations
compare favorably with direct Monte Carlo simulations. We consider the following applications: an intermittently unstable mechanical oscillator excited by correlated noise;
a complex mode in a turbulent signal with fixed frequency, where nonlinear mode
interaction terms are replaced by a stochastic drag and additive white noise forcing;
and a stochastic Mathieu equation, featuring intermittent parametric resonance.
Thesis Supervisor: Themistoklis P. Sapsis
Title: Assistant Professor of Mechanical Engineering
iii
Acknowledgments
First and foremost, I would like to thank my supervisor Professor Themistoklis Sapsis
for his invaluable guidance and encouragement. I have learned a tremendous amount
from his supervision and this thesis would not have happened without his support.
I would like to acknowledge the Naval Engineering Education Center (NEEC
3002883706), the Office of Naval Research (ONR N00014-14-1-0520) for supporting
this research, and the Martin A. Abkowitz fund for a travel award. I thank the
American Bureau of Shipping for gracious support through a fellowship and also
support through a Pappalardo Fellowship made possible by A. Neil Pappalardo.
I thank all my friends here at MIT and around Cambridge and Boston, they have
made these last two years a pleasant and joyful experience. Last but not least, I thank
my parents and two sisters for their unconditional support and encouragement, no
words can describe how much gratitude I owe them.
iv
Contents
1
Introduction
1
2
Probabilistic description of intermittent systems
5
Problem statement and formulation . . . . . . . . . . . . .
5
2.2
Quantification of the stable regime
. . . . . . . . . . . . .
8
2.3
Quantification of the unstable regime . . . . . . . . . . . .
9
2.3.1
Stochastic description of the growth phase . . . . .
9
2.3.2
Stochastic description of the decay phase . . . . . .
10
Probability of the stable and unstable regimes . . . . . . .
11
.
.
.
.
Instabilities driven by Gaussian processes
13
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3.2
Average time below and above the zero level . . . . . . . .
14
3.3
Distribution of time below the zero level
. . . . . . . . . .
15
.
.
3.1
.
3
.
2.4
.
2.1
4 Application to a parametrically excited oscillator
17
Problem formulation . . . . . . . . . . . . . . . . . . . . .
. . . . .
18
4.2
Probability distribution in the stable regime . . . . . . . .
. . . . .
19
4.3
Probability distribution in the unstable regime . . . . . . .
. . . . .
20
Asymptotics for the unstable probability distribution . . . . .
23
.
.
.
4.1
4.3.1
Envelope probability distribution in the stable regime . . .
. . . . .
26
4.5
Probability distribution for the velocity . . . . . . . . . . .
. . . . .
27
4.6
Summary of the analytical results . . . . . . . . . . . . . .
.
.
.
4.4
v
29
4.7
5
6
. . . . . . . .
30
4.7.1
Comparison with Monte Carlo simulations . . . . . . . . . . .
32
4.7.2
Parameters and region of validity . . . . .... . . . . . . . . .
34
Discussion and comparison with numerical simulations
Application to intermittently unstable turbulent modes
39
5.1
Probability distribution in the stable regime . . . . . . . . . . . . . .
41
5.2
Probability distribution in the unstable regime . . . . . . . . . . . . .
42
5.3
Summary of the analytical results . . . . . . . . . . . . . . . . . . . .
44
5.4
Discussion and comparison with numerical simulations
. . . . . . . .
44
47
Application to the stochastic Mathieu equation
6.1
Problem definition and formulation . . . . . . . . . . . . . . . . . . .
49
Derivation of the slowly varying variables . . . . . . . . . . . .
51
. . . . . . . . . . . . .
53
6.1.1
6.2
Probability distribution for the slow variables
6.2.1
Stable regime distribution
. . . . . . . . . . . . . . . . . . . .
54
6.2.2
Unstable regime distribution . . . . . . . . . . . . . . . . . . .
54
6.3
Summary of the analytical results . . . . . . . . . . . . . . . . . . . .
56
6.4
Discussion and comparison with numerical simulations
. . . . . . . .
56
Conclusions and future work
59
A Gaussian Process Simulation
61
7
vi
List of Figures
2-1
An intermittently unstable system. The system (top) experiences a
large magnitude response when the parametric excitation process (bottom) crosses the zero level. Dashed green lines denote the envelope
of the process during the stable regime. The transient instability can
be described by two stages, an exponential growth phase and then a
subsequent decay phase that relaxes the response back to the stable
regim e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-1
8
Comparison of the analytic distribution PT(t) (3.6) with Monte Carlo
simulation, generated using 1000 ensembles of a Gaussian process with
correlation R(T) = exp(-r 2 /2) from to
=
0 to tf = 1000, for two
different sets of parameters: m = 5.0, k
k = 2.4 (right).
4-1
1.6 (left) and m = 5.0,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
Scatter plot and contours of the joint pdf of T and A, for r= -2.08
(m = 5.00, k
=
2.40). Samples for A are drawn by inverse-sampling
the cumulative distribution function and samples of T are drawn from
realizations of a Gaussian process using 1000 ensembles of time length
t
4-2
=
1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exact integral in equation (4.15) compared to the asymptotic expansion
given in equation (4.25), for a fixed uo and m
4-3
23
=
5.00, k = 2.00. .....
25
For the computation of the velocity pdf, we approximate the envelope
of the process during the extreme event with a harmonic function with
consistent period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
28
4-4 Sample realization (top) and the corresponding parametric excitation
process (bottom) demonstrating the typical system response during an
intermittent event for position (in thick blue) and velocity (in thin red)
for ri = -2.27 (k = 2.2, m = 5.0), c = 0.53. . . . . . . . . . . . . . . .
4-5
31
Typical superposition of the stable (in dashed blue) and unstable (in
dotted red) part of the total probability law decomposition (2.6) of the
full analytical pdf.
4-6
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
Demonstration of the cusp phenomenon, which is prominent in regimes
with very frequent instabilities and/or instabilities of long duration, due
to approximations made in the unstable regime. . . . . . . . . . . . .
33
4-7 Analytic pdfs (4.41), (4.45) compared with results from Monte Carlo
simulations for position (left) and velocity (right), for parameters: k =
1.8, m = 5.0, c = 0.38, a. = 0.75.
4-8
. . . . . . . . . . . . . . . . . . . .
34
Analytic pdfs (4.41), (4.45) compared with results from Monte Carlo
simulations for position (left) and velocity (right), for parameters: k =
2.2, m = 5.0, c = 0.53, o, = 0.75.
4-9
. . . . . . . . . . . . . . . . . . . .
34
Analytic pdfs (4.41), (4.45) compared with results from Monte Carlo
simulations for position (left) and velocity (right), for parameters: k
2.6, m = 5.0, c = 0.69, a, = 0.75.
=
. . . . . . . . . . . . . . . . . . . .
35
4-10 KL divergence for position (left) and velocity (right) between analytic
and Monte Carlo results for various values of damping (in terms of the
number of oscillations N after an instability) and k, which controls the
number of instabilities for a fixed m. For fixed m = 5.0. . . . . . . . .
5-1
Regime 1: Sample path (left) and analytic pdf (5.15) compared with
results from Monte Carlo simulations (right), for parameters: w
a = 0.5, m = 2.25, k = 2.0, L = 0.125.
5-2
37
=
1.78,
. . . . . . . . . . . . . . . . .
45
Regime 2: Sample path (left) and analytic pdf (5.15) compared with
results from Monte Carlo simulations (right), for parameters: w = 1.78,
a = 0.1, m. = 0.55, k = 0.50, L = 2.0. . . . . . . . . . . . . . . . . . .
viii
46
5-3
Regime 3: Sample path (left) and analytic pdf (5.15) compared with
results from Monte Carlo simulations (right), for parameters: w = 1.78,
a- 0.25, m = 8.1, k = 1.414, L = 4.0.
6-1
. . . . . . . . . . . . . . . . .
Oscillatory growth due to parametric resonance (left) for 6 > 0 and
exponential growth (right) when 6 < 0 of the Mathieu equation. . . .
6-2
46
48
Ince-Strutt diagram showing the classical resonance tongues (shaded)
for particular combinations of c and 6, for the Mathieu equation (6.1)
non-dimensionalized by Q. Transition frequencies at 6 = w /Q2
(n/2)2 . Stability boundaries are computed via Hill's infinite deteriminant [1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-3
Sample realization of the Mathieu equation (6.4) (top). The random
amplitude forcing term
#(t)
(bottom) triggers intermittent resonance
when it crosses above or below the instability threshold (dashed lines).
6-4
49
50
Comparison of Monte-Carlo results (6.25) (red curve) and analytic
results (6.24) of the Mathieu equation for various intermittency levels
with 6-4a being least intermittent and 6-4f most intermittent. ....
ix
58
Chapter 1
Introduction
A wide range of dynamical systems describing physical and technological processes
are characterized by intermittency. In other words, the property of having sporadic
responses of extreme magnitude when the system is "pushed" away from its statistical
equilibrium. This intermittent response is usually formulated through the interplay
of stochastic excitation, which can trigger internal system instabilities, deterministic
restoring forces (usually in terms of a potential) and dissipation terms. The presence of
intermittent instabilities can be identified by large amplitude spikes in the time series
of the system or heavy-tails in the probability density function (pdf) of the response.
It is often the case that, despite the high dimensionality of the stable attractor, where
the system resides most of the time, an extreme response of short duration is due
to an intermittent instability occurring over a single mode. This scenario does not
exclude the case of having more than one intermittent mode, as long as the extreme
responses of these modes are statistically independent. For this case, it may be possible
to analytically approximate the probabilistic structure of these modes and understand
the effect of the unstable dynamics on the heavy-tails of the response.
Instabilities of this kind are common in dynamical systems with uncertainty. One
of the most popular examples is modes in turbulent fluid flows and nonlinear water
waves subjected to nonlinear energy exchanges that occur in an intermittent fashion
and result in intermittent responses [2, 3, 4, 5, 6, 7, 8]. Prototype systems that mimic
these properties were introduced in [9, 10, 11]. In recent works, it has been shown
1
that properly designed single mode models can describe intermittent responses, even
in very complex systems characterized by high dimensional attractors [11, 12, 13].
Another broad class of such systems includes mechanical configurations subjected to
parametric excitations, such as the parametric resonance of oscillators and ship rolling
motion [1, 14, 15, 16, 17, 18]. Finally, intermittency can often be found in nonlinear
systems that contain an invariant manifold that locally loses its transverse stability
properties, e.g. in the motion of finite-size particles in fluids, but also in biological
and mechanical systems with slow-fast dynamics [19, 20, 21, 22].
In all of these systems, the complexity of the unstable dynamics is often combined
with stochasticity introduced by persistent instabilities that lead to chaotic dynamics,
as well as by the random characteristics of external excitations. The structure of the
stochasticity introduced by these factors plays an important role on the underlying
dynamics of the system response. In particular, for a typical case the stochastic
excitation is colored noise, i.e. noise with a finite correlation time length. Due to
the possibility of large excursions by the colored stochastic excitation from its mean
value into an "unsafe-region," where instabilities are triggered, extreme events may be
particularly severe. Therefore, for an accurate description of the probabilistic system
dynamics, it is essential to develop analytical methods that will be able to accurately
capture the effects of this correlation in the excitation processes for intermittent modes.
However, analytical modeling in this case is particularly difficult, since even for lowdimensional systems, standard methods that describe the pdf of the response (such
as the Fokker-Planck equation) are not available or computationally too expensive.
For globally stable dynamical systems numerous techniques have been developed
to analyze extreme responses (see e.g. [23, 24, 25]). Various steps are involved that lead
to elegant and useful results, however the starting point is usually the assumption of
stationarity in the system response and this is not the case for intermittently unstable
systems. Extreme value theory [26, 27, 28, 29, 30] is also a widely applied method
that focuses on thoroughly analyzing the extreme properties of stationary stochastic
processes following various distributions. In this case, however, the analysis does not
2
take into account system dynamics and is usually restricted to very specific forms of
correlation functions [26, 27].
Approaches that place more emphasis on the dynamics include averaging of the
governing equations and variations thereof. However, the inherently transient nature
of intermittent instabilities makes it impossible for averaging techniques to capture
their effect on the response statistics. Other modeling attempts include approximation
of the correlated parametric stochastic excitation by white noise (see, for example
[31]
for an application to parametric ship rolling motion). However, even though this
idealization considerably simplifies the analysis and leads to analytical results, it
underestimates the intensity of extreme events, since instabilities do not have enough
time to develop (i.e., the system spends an infinitesimal time in the unstable regime,
which is insufficient for the system to depart from the stable attractor); in other words,
the original system, with colored excitation, has stronger more intense instabilities,
which a white noise idealization would fail to capture. For this case, heavy-tails can
be observed only as long as the parametric excitation is very intense, which for many
systems is an unrealistic condition.
In this thesis our goal is to describe and develop a fundamentally novel method
that will allow for the analytical approximation of the probability distributionfunction
(pdf) of modes associated with intermittent instabilities and extreme responses due to
parametric excitation by colored noise. This analytic approach will provide a direct link
between dynamics and response statistics in the presence of intermittent instabilities.
We decompose the problem by conditioning the system response on the occurrence
of unstable and stable dynamics through a total probability law argument. This idea
enables the separate analysis of the system response in the two regimes and allows us
to accurately capture the heavy-tails that arise due to intermittent instabilities. The
full probability distribution of the system state is then reconstructed by combining
results from the two regimes. We illustrate this approach on several prototype systems
with nontrivial statistics that arise in problems of turbulent flows, nonlinear waves,
and mechanics. We will thoroughly examine the extent of validity of the derived
approximations by direct comparison with numerical Monte Carlo simulations.
3
SL.
..
.
-
We emphasize that the new method we describe here offers vast computational
savings and accurate tail statistics. These two qualities, computational speed and
accurate tail statistics, are typically at odds with each other when we look at other
methods. For example, Fokker-Planck based methods become prohibitively expensive
when we are interested in tail statistics and require very accurate numerical methods
(since tail events have extremely low probabilities) to solve the resulting, usually high
dimensional, differential equations. The method described in this thesis overcomes
this contradictory condition (fast algorithm vs. accurate tails) by recognizing that tail
events are governed by different dynamics than the dynamics of the typical response.
Accordingly, we decompose the statistics in light of the fact that different dynamics
govern the two regimes, stable and unstable, and make appropriate approximations in
the two regimes to derive the response pdf for the system we are interested in. Here,
we are interested in deriving analytical results for the equilibrium pdf, and thus the
computational savings can be several orders of magnitude when we compare them
with, say, Monte Carlo results, which is the most accurate but the most expensive
technique.
4
Chapter 2
Probabilistic description of
intermittent systems
In this chapter we formulate our method for describing the probabilistic structure
of the response pdf for intermittently unstable dynamical systems. We begin from
a general dynamical system describing modes subjected to intermittent instabilities
and show how to apply the idea of a probabilistic decomposition by conditioning on
the two relevant dynamical regimes (stable and unstable regimes) through a total
probability law argument. This requires several components including determining
the conditional response pdf in the two regimes and also the probability of being in
the two regimes; we show to determine these quantities in an analytical framework.
The results from this chapter are the foundation for the applications we present in
later chapters.
2.1
Problem statement and formulation
Let (E, B, P) be a probability space, where
E is the sample space
with 6 E E denoting
an elementary event of the sample space, B is the associated --algebra of the sample
space, and P is a probability measure. We denote by Px the probability measure and
by Px the corresponding probability density function, if appropriate, of a random
quantity X. We are interested in describing the statistical characteristics of modes
5
subjected to intermittent instabilities. We consider a general dynamical system
=
G(x, t),
x ER ,
(2.1)
and assume that its response is an ergodic stochastic process. The analysis will rely
on the following additional assumptions related to the form of the intermittent instabilities:
Al The instabilities are rare enough that they can be considered statistically independent and have finite duration (localized).
A2 During an extreme event the influenced modes have decoupled dynamics. Moreover, for each one of these modes, the instability is the governing mechanism; i.e.,
nonlinear terms are not considered to be important during these fast transitions;
they are only considered to be important during the stable dynamics.
A3 After each extreme event there is a relaxation phase that brings the system back
to its stable stochastic attractor.
Under these assumptions we may express each intermittent mode, denoted by u(t; 0) E
R, where 0 E
e
(for notational simplicity, we drop explicit dependence on the proba-
bility space for random processes whenever clear), as a dynamical system of the form
,it + a(t; )u)+ E((U, v) = Ec(t; 0),
(2.2)
where F > 0 is a small quantity and ((u, v) is a nonlinear term (nonlinearizable), which
may also depend on other system variables v E R"'. Since E is a small quantity, we
can assume that the nonlinear term is important only in the stable regime. The
stochastic processes a(t; 0) and (t; 0) are assumed stationary with known statistical
characteristics. For a(t) we will make the additional assumption that its statistical
mean is positive a > 0 so that the above system has a stable attractor (we denote the
mean value operator, the ensemble average, by an overbar E). The above equation
can be thought of as describing the modulation envelope for a complex mode with a
6
narrowband spectrum, or one of the coordinates that describe the transverse dynamics
to an invariant slow manifold. As motivation, the model (2.2) can be thought as an
expansion of (2.1) in the form
,J= G((x2,
.,XN
-3)
Zaijxj + E bijXj, +
j,k
j
(aii + biix1 + b+X2 +
(2.4)
.
)xb+
+abx + ++ -bikXx
- , (2.5)
where we model excitations from other system variables as external noise ((t; 0),
quadratic interactions as parametric noise a(t; 0), and nonlinear terms are denoted
by(
The objective in this thesis is to derive analytical approximations for the probability distribution function of the system response, taking into account intermittent
instabilities that arise due to the effect of the stochastic process a(t). In particular,
these instabilities are triggered when a(t) < 0 and force the system to depart from
the stable attractor. Therefore, the system has two regimes where the underlying dynamics behave differently: the stable regime where a(t) > 0 and the unstable regime
that is triggered when a(t) < 0 (Figure 2-1). Motivated by this behavior, we quantify
the system's response by conditioning the probability distribution of the response on
stable regimes and unstable events, and reconstruct the full distribution according to
the law of total probability,
Ps(x) = Pa(x I stable regime)P (stable regime)+
Pu(x I unstable regime)P(unstable regime),
(2.6)
thereby separating the two regions of interest so that they can be individually studied.
The method we employ relies on the derivation of the probability distribution for each
of the terms in (2.6) and then the reconstruction of the full distribution of the system
by the total probability law. This idea allows us to capture the dramatically different
statistics that govern the system response in the two regimes. Essentially, we decouple
the response pdf into two parts: a probability density function with rapidly decaying
tails (typically Gaussian) and a heavy-tail distribution with very low probability close
to zero. In the following sections, we provide a description of the method involved for
the statistical determination of the terms involved in (2.6).
0
50
100
t
150
200
(t;)
15
instability triggered
5
0
0
50
100
150
200
Figure 2-1: An intermittently unstable system. The system (top) experiences a large
magnitude response when the parametric excitation process (bottom) crosses the zero
level. Dashed green lines denote the envelope of the process during the stable regime.
The transient instability can be described by two stages, an exponential growth phase
and then a subsequent decay phase that relaxes the response back to the stable regime.
2.2
Quantification of the stable regime
During the stable reginme we have by definition a > 0; therefore, under this condition
the considered mode is stable. Note that this condition is also true during the relaxation
(decay) phase after an extreme event when a switches to positive values a > 0, but
is not yet relaxed back to the stable attractor. To this end, we cannot directly relate
the duration of being on the stable attractor with the probability
(a > 0), but a
correction should be made. We present this correction later in Section 2.3.2. Here
we focus on characterizing the probability density function of the system under the
assumption that it has relaxed to the stable attractor, and, moreover, we have a > 0.
8
As a first-order estimate of the stable dynamics, we approximate the original
dynamical system for the intermittent mode by the stable system
t + dja>OU + &((u,v) =l(t;
0),
(2.7)
where MaI>o denotes the conditional average of the process a(t) given that this is
positive. The determination of the statistical structure of the stable attractor for (2.7)
can be done with a variety of analytical and numerical methods, such as the FokkerPlanck equation if the process
(t) is white noise (see, e.g., [32, 24]) or the joint
response-excitation equations otherwise [33, 34]. Using one of these methods, we can
obtain the probability distribution function for the statistical steady state of the
system, that is P (x Istable regime).
2.3
2.3.1
Quantification of the unstable regime
Stochastic description of the growth phase
In contrast to the stable regime, the unstable regime is far more complicated due
to its inherently transient nature. In addition, the unstable regime consists of two
distinct phases: a growth phase where a < 0 with positive Lyapunov exponent, and a
subsequent decay phase that starts when a has a zero upcrossing and is characterized
by a > 0 with negative Lyapunov exponent. We first consider the growth phase, where
we rely on assumption A2, according to which the dominant mechanism is the term
related to the instability. Under this assumption, the first-order approximation of the
dynamics during the growth phase is given by
,it + a(t; 0)u = 0,
(2.8)
where no is a random initial condition described by the probability measure in the
stable regime, T is the random duration of the downcrossing event a < 0, and A is
the random growth exponent, which for each extreme event, due to the rapid nature
9
of the growth phase, can be approximated by
A ~
a(t; 0) dt,
(2.9)
where t* represents the time of the start of the instability and r the duration that
a < 0. This implies the following representation for instabilities during the growth
phase
'u(t; 9) = uoeT,
(2.10)
and therefore
P(U > U* I a < 0) = P(UoeAT > u* I a < 0)
= P(uoeAT > U* I a < 0, uo) P(uo),
(2.11)
where u* is the value of the response we are interested in. The right-hand side of (2.11)
is a derived distribution depending on the probabilistic structure of A and T. The
initial value uO is a random variable with statistical characteristics corresponding to
the stable regime of the system, in other words by P(u I stable regime). Hence, to
determine the required probability distribution we need only to know P(a, T Ia < 0),
i.e., the joint probability distribution function of a (given that this is negative) and
the duration of the time interval over which a is negative. This distribution involves
only the excitation process a, and for the Gaussian case it can be approximated
analytically (see Section 3). Alternatively, one can compute this distribution using
numerically generated random realizations that respect the statistical characteristics
of the process.
2.3.2
Stochastic description of the decay phase
The decay phase is also an inherently transient stage. It occurs right after the growth
phase of an instability, when a has an upcrossing of the zero level, and is therefore
characterized by positive values of a, and drives the system back to the stable attractor.
10
6
9
1
--
Au6vw
To provide a statistical description of the relaxation phase, we first note the strong
connection between the growth and decay phases. In particular, as shown in Figure 21, for each extreme event there is a one-to-one correspondence for the values of the
intermittent variable a between the growth phase and the decay phase. By focusing
on an individual extreme event, we note that the probability of u exceeding a certain
threshold during the growth phase is equal with the probability of u exceeding the
same threshold during the decay phase. Thus over the total instability we have
P(n > u* I unstable regime) =
P(u > u* I instability - decay) = P(u > u* I instability - growth),
(2.12)
where the conditional distribution for the growth phase has been determined in (2.11).
2.4
Probability of the stable and unstable regimes
In the final step we determine the relative duration of the stable and unstable regimes.
To do so, we rely on the ergodicity of the system response. This allows us to quantify
the probability of a stable and an unstable event in terms of their temporal durations.
The probability of having an instability is not simply P(a < 0), because of the decay
phase the duration of an instability will be longer than the duration of the event a < 0.
To determine the typical duration of the decay phase, we first note that during the
growth phase we have
Up =
oe"I <OT.<,
(2.13)
where Ta<o is the duration for which a < 0 and up is the peak value of u during the
instability. Similarly, for the decay phase we utilize system (2.7) and obtain
nO=
upe-Zila>oTdecay
11
(2.14)
......................
...............
Combining the last two equations (2.13) and (2.14), we have
Ta<o
-610y>0
Tdecay
dIO<O
(2.15)
The equation above (2.15) expresses the typical ratio between the growth and the
decay phases. Thus, the total duration of an unstable event is given by the sum of
the duration of these two phases
Tinst
(I -
(2.16)
l'>O) Ta<O.
Using this result, we can express the total probability of being in an unstable regime
by
P(unstable regime) = (1 -
I'>O) P(a < 0).
(2.17)
Note that since we have assumed in Al that instabilities are sufficiently rare so that
instabilities do not overlap and that instabilities are statistically independent, we will
always have P(unstable regime)
< 1. Hence, the corresponding probability of being
in the stable regime is
P(stable regime) =1 - (I - T ">)
12
P(a < 0).
(2.18)
Chapter 3
Instabilities driven by Gaussian
processes
In this chapter we recall relevant statistical properties associated with Gaussian processes. First, we define what is a Gaussian processes, and then describe important
statistical quantities associated with a Gaussian excitation mechanism; that is, systems where instabilities are driven by Gaussian processes. In particular, we note the
importance of the zero-level of a(t) in the dynamical system (2.2). This threshold level
is critical since it defines the boundary between a stable and an unstable response. We
quantify the following statistical properties associated with the zero-level of a Gaussian excitation process: the probability that a(t) is above and below the zero-level,
the average duration spent below the zero-level, and the probability distribution of
the length of time intervals spent below the zero-level. These important quantities
will be used in future chapters, where we assume a stationary Gaussian excitation
mechanism.
3.1
Introduction
A stochastic process is said to be Gaussian if all of its finite-dimensional distributions
are normal. In other words X(t), where t E R is said to be Gaussian if for any choice
13
of n and t1, . .
,t
E R we have
X
(3.1)
X.)~ N(p, E),
(X,.
for some expectation vector ti and covariance matrix E. Gaussian process are completely determined by their expectation function p(t) = E(X(t)) and covariance
function R(s, t) = Cov(X(s), X(t)). Furthermore, a Gaussian process is said to be
stationary if its expectation function p(t) is constant and its covariance function is
invariant under translations in time, i.e. R(T) = Cov(X(t), X(t +
T)).
In addition,
an important property associated with R(r) is its relation with the so-called power
spectral density function of X(t),
Rx(T)
J
Sy(w)e
' dw.
(3.2)
-00
By the Wiener-Khintchine theorem, the functions R(r) and S(w) form a Fourier transform pair. Algorithms for simulating Gaussian processes can be found in Appendix A.
We consider the case where a(t) is a stationary Gaussian process with mean m and
variance k 2 , and for convenience write a(t)
m. + ky(t), so that -y(t) has zero mean
and unit variance. Therefore, the rare event threshold, in terms of the process -(t), is
given by i = -mn/k.
We assume that second order properties, such as the correlation
function or power spectral density function, for a(t) are known. In addition, we denote
by 0 ( -) the standard normal probability density function and by <D( -) the standard
normal cumulative probability density function. Since 7(t) is a stationary Gaussian
process, the probability that the stochastic process is in the two states IP(a < 0) and
IP(a > 0) is ]P(- <'1)
3.2
=
41(q) and P(' >
j)
1 - 4b(q), respectively.
Average time below and above the zero level
Here we determine the average length of the intervals that a(t) spends above and
below the zero level. For the case a(t) < 0, that is y(t) < i, the expected number of
14
1
11,1111"'."11111,11,1
....
............. I, .....
upcrossings of this threshold per unit time is given by Rice's formula [35, 36] (for a
stochastic process with unit variance)
-R
)uP ( , ') d"(
0( ) exp(-4 2/2),
(3.3)
0
where N+ (ij) is the average number of up crossings of level q per unit time, which is
equivalent to the average number of downcrossings N (,j), Py is the joint pdf of y
and its time derivative
, and the primes denote differentiation. The expected number
of crossings is finite if and only if -y(t) has a finite second spectral moment [36].
The average length of the interval that 7(t) spends below the threshold
j
can then
be determined by noting that this probability is given by the product of the number
of downcrossings of the threshold per unit time and the average length of the intervals
for which 'y(t) is below the threshold q [37]
- P( < q)
(q)
Tao =
(3.4)
Hence, using the result (3.3), we have
Ta<00) = 27r exp(?2 /2)<()
"(0)
3.3
(3.5)
Distribution of time below the zero level
Here we recall a result regarding the distribution of time that the stochastic process
7(t) spends below the threshold level r, that is the probability of the length of intervals
for which -y(t) < q.
In general, it is not possible to derive an exact analytical expression for the distribution of time intervals given 'y(t) < q, in other words the distribution of the length of
time between a downcrossing and an upcrossing. However, the asymptotic expression
in the limit il -+ -oo is given by [37] (henceforth we denote Ta<o by T for simplicity)
15
-
wt
PT(t) -T 2 exP(-wt2/4T ),
2T
(3.6)
2T2 /ir, where T is given
which is a Rayleigh distribution with scale parameter
by (3.5). This approximation is valid for small interval lengths t, since the derivation
assumes that when a downcrossing occurs at time t1 only a single upcrossing occurs
at time t2
= t1
+ t, neglecting the possibility of multiple crossings in between the
two time instances. In Figure 3-1 the analytic distribution (3.6) is compared with
numerical results, and as expected, we see that the analytic expression agrees well
with the numerical results for small interval lengths, where the likelihood of multiple
upcrossings in the interval is small. In addition, note that the only relevant parameter
of the stochastic process -y(t) that controls the scale of the distribution in equation (3.6)
is the mean time spent below the threshold level, which depends only on the threshold
rj and R"(0).
101
101
- -
Monte Carlo
10
10
10
10
10o2
10 -2
10
10
0
10
numeric mean 0.744
analytic mean = 0.737
1
0
1N0
-4
10-41
1
-Analytic
- Monte Carlo
-Analytic
2
3
t
4
5
6
10 0
7
numeric mean 1.03
analytic mean = 1.02
1
2
3
4
5
6
Figure 3-1: Comparison of the analytic distribution PT(t) (3.6) with Monte Carlo
simulation, generated using 1000 ensembles of a Gaussian process with correlation
R(r)
m
=
exp(-r 2 /2) from to = 0 to tf = 1000, for two different sets of parameters:
5.0, k = 1.6 (left) and m = 5.0, k = 2.4 (right).
=
16
7
Chapter 4
Application to a parametrically
excited oscillator
In this chapter we consider parametrically excited motions. In parametrically excited
systems, excitations appear in the governing differential equations as time varying
coefficients. We are interested in the realistic case of stochastic or random parametric excitations, which can lead to transient instabilities or transient resonances. We
consider systems described by
X +
cx +
K(t)X
=
(t),
(4.1)
where K(t) is in general a time correlated stochastic parametric excitation process,
i.e. colored noise, and
(t) is additive noise, which may also be time correlated. In
Chapter 6 we discuss the intermittent resonance case, whereby even a small amplitude
parametric excitation can produce a large response. Here we focus on parametric
excitations the lead to transient instabilities. In this case, intermittent instabilities
are triggered due to large excursions of the parametric excitation process K(t) into the
region K < 0. We consider this as a rare problem, namely that these large excursions
occur with relatively low probability, and consequently the instabilities they trigger
are rare events, so that on average - > 0 and the response is stable.
17
Parametrically excited systems arise in many areas of engineering and physics.
We mention but a few examples of physical systems where the governing differential
equation is parametrically excited (see e.g. [38, 39, 1, 24] for additional examples and
further details), they include: the transverse motion of elastic beams excited axially;
electrical circuits, such as LC circuits (inductor connected to a capacitor in series)
with time varying capacitance [40]; the motion of pendulums, including cases where
the support is excited or the length of the rod is time varying; the transverse vibration
of a taut strings undergoing variations in tension or torsion [41].
A particularly important example in ocean engineering is ship roll motion in the
presence of random waves [31, 17, 423. Here random restoring forces, due to the action
of irregular water waves, show up as parametric terms in simplified models describing the ship's roll response. The possibility of indirectly excited large-amplitude roll
motions are a serious threat to the safety of a ship and human life. This requires
understanding the action of stochastic terms and nonlinearity in the governing equations of motion. Several models and methods have been proposed in connection with
the problem to predict large roll motions and capsizing probabilities, where we are
pressed by the need for easy to use analytical results and inexpensive computational
methods [43, 44, 45, 46, 47]. Nevertheless many deficiencies in various methods and
models remain owing to the challenges associated with capturing the effects of the
correlated nature of real ocean waves and system nonlinearities and, most importantly,
quantifying the resulting non-Gaussian response statistics. The results here (see also
Chapter 6) offer a novel strategy for efficiently quantifying large roll motions in an
analytical setting, and is able to capture the effect of the correlated nature of real
ocean waves.
4.1
Problem formulation
With the presentation of the method in Chapter 2, we now proceed to our first application. We consider the following single-degree-of-freedom oscillator, under parametric
18
..............
stochastic excitation and additive white noise forcing,
z(t) + czt(t) +
ii(t;
)x(t) = oXW(t; 0),
(4.2)
where c is damping, K(t) is a stationary Gaussian process of a given power spectrum
SK(w) or correlation function R,(r) with mean m and variance k 2 , and W(t) is white
noise with amplitude or. Here 77
--m/k defines the extreme event threshold. We
can alternatively write equation (4.2) in state space form as
dx 1
(4.3)
=x2 dt
dx 2 = -(cx2 +
K(t)x1)
dt
+ o-x
(4.4)
dW(t).
The presentation will follow Chapter 2, although the analytical expressions for the
pdf is computed directly for position and velocity (instead of the envelope process
u) for instructive purposes. At the end of this chapter we present comparisons with
direct numerical simulations and thoroughly examine the limits of validity for the
approximations we have made.
4.2
Probability distribution in the stable regime
We first derive the probability distribution for (4.2) given that the system is in the
o. This
stable regime. Following Section 2.2, we replace K(t) by the mean W 2
K
approximation is valid since only small fluctuations of the process
around the
K(t)
mean w occur in this state, and these fluctuations have a minor impact on the system's
probabilistic response. Making this approximation, (4.3) becomes
:z(t) + cg(t) + w2x(t) = Uo-W(t).
(4.5)
To determine the mean w., note that its probability density function is given by
P, (x I r
> 0)
k (I 19
k
<(())
(4.6)
..............
and thus
2
WS
Re~
m+ k
OW___
.)n
(4.7)
The density in the stable regime can now be found by seeking a stationary solution
of the Fokker-Planck equation of the form P(xI, X2) = P(H), where H =
(W x + x2)
is the total mechanical energy of the system. In this case the Fokker-Planck equation
simplifies to [24]
1 dP
=0
cP(H)+-o
2 dH
> P(H)=qexp
2
2 H
ar
(4.8)
where q = cw,/(7ro-x) is a normalization constant. Taking the marginal with respect
to X2 gives the probability density for the system's position
P.,(x I stable regime) =
which is Gaussian with variance o 2 =
a/
exp
.5
-x
2
(4.9)
(2cW2). Similarly, taking the marginal with
respect to x, gives the probability density for the system's velocity
P, (x Istable regime) =
cc
2 exPy
2 2
,
(4.10)
where the variance is o 2 = o /(2c).
4.3
Probability distribution in the unstable regime
Here we derive the probability distribution for the system's position x, during the
growth stage of an instability. In the unstable regime, the system initially undergoes
exponential growth due to the stochastic process ri(t) crossing below the zero level,
which is the mechanism that triggers the instability. After a finite duration of exponential growth, which stops when K(t) crosses above the zero level, and due to the
large velocity gradients that result, friction damps the response back to the stable
attractor. Following assumption A2, we consider the parametric excitation as the
20
primary mechanism driving the instability during the growth phase, and therefore
ignore the effect of friction during this stage of the instability.
We treat the system response as a narrowbandprocess. This means that the response
has a spectrum that is narrowly distributed around a mean frequency, i.e., q = u-,/w. <
1, where w, = f wS(w) dw and o-
f(w
-
w,) 2S(w) dw [48]. We can therefore describe
the instability in terms of the envelope u for the position variable by averaging over the
fast frequency w,. During the growth phase we have u ~ uoeAT where uo is the random
value of the position's envelope before an instability (to be determined in Section 4.4),
T is the random duration of the event s < 0, and A is the random growth exponent,
which will be the same for both the position and the velocity variables. Substituting
this representation into (4.2), we find A2 + ~ 0,- so that the positive eigenvalue is
given by A ~
-
.
To proceed, we set u = uoeAT and y = A; then, from a change of
variables,
Psy(u, y) = PAT(A, t) Idet[0(A, t)/0(u, y)]
,
(4.11)
where we understand that the pdf is conditional upon K < 0 and uO. Next we assume
that T and A are independent (see Figure 4-1) and thus we have
PA(A)P(t)
Py (U, Y) = PA(At)
uO Aexp (At)
_____/__
_1
uy
PA (Y)PT (
u > iUo, y > 0.
y
(4.12)
Taking the marginal density gives
Pu(u) =- J-PA(y)P T (lo(/UO)) dy.
U I0 y
Now, since P(A) = P(\
PA(A)
-
(4.13)
y
< 0) from another change of variables we obtain
-l,
2A
<41r0)
2)
P(-A2( -
A 2 + in
(4.14)
k
k~lb(,q) (
2A
) A>0,
since P(K.) is normally distributed. Combining this result and the fact that PT is given
by the Rayleigh distribution (3.6), with (4.13) gives the final result for the probability
21
....... MMMMM-'-"-
density function
I K < Oo)
y2 +m)
7log(u/uo) 71
kT
<(r)
k
I0 y
x exp
-
I2
4T2y2
log(U/uo)2
dy,
It > Ito,
(4.15)
where T is given in (3.5). Moreover, utilizing the expression for the duration of a
typical extreme event (2.16), we have
Tinst
1
(4.16)
+ 2XAo TK<O,
where for the relaxation phase we have the rate of decay for the envelope being -c/2.
To formulate our result in terms of the position variable, we refer to the narrowband
approximation made at the beginning of this analysis. This will give approximately
x, = I cos W, where
'p
is a uniform random variable distributed between 0 and 27r.
The probability density function for z = cos ' is given by Pz(x) = 1/(7/w
XC
1
-
x2),
[-1, 1]. To avoid double integrals for the computation of the pdf for the position
variable, we approximate the pdf for z by P2 ( x)
=
-(6(x + 1) + 6(x - 1)). This will
give the following approximation for the conditionally unstable pdf:
1
P,(x I K < 0, uo) = -P(Ix I K, < 0, uo).
2
(4.17)
We note that as presented in Section 2, the pdf for the envelope u may also be
used for the decay phase. However, during the decay phase the oscillatory character
(with frequency w,) of the response has to also be taken into account. Nevertheless, to
keep expressions simple we will use equation (4.17) to describe the pdf for the position
in the decay phase as well. We will show that this approximation compares favorably
with numerical simulations.
22
-
3
2.5
2,
1
-
0.50
0
6
5
4
3
T
2
1
-2.08
Figure 4-1: Scatter plot and contours of the joint pdf of T and A, for r=
cumulative
(M = 5.00, k = 2.40). Samples for A are drawn by inverse-sampling the
distribution function and samples of T are drawn from realizations of a Gaussian process
using 1000 ensembles of time length t = 1000.
4.3.1
Asymptotics for the unstable probability distribution
In the spirit of keeping our final expressions simple, here we derive an asymptotic
expansion for the integral (4.15) that describes the distribution for the position's
envelope during the growth phase of an instability, as u -4 oo, treating 1 0 as a
parameter. For convenience in (4.15) let PL(tu
7 log(a/ao)
((u, y; uo) =2
v/2wkT <b(rj)uy
7r log(a//uo)
=
2wkt2
V 27kT
(
exp
x < 0, < o) = Jo (u, y; 0o) dy, that is,
(y 2 + r)
2
~
2k
2
7r
-2
4T y
2
o1y/ao)
(4.18)
(4.19)
exp(O(u, y; 'uo)),
u{)
2
<b(,q),ty
where O(u, y; ao) denotes the terms being exponentiated (we suppress the explicit
dependence on the parameter
0o
,-No
P
,(
<0,ao) /(U, Y; -o) dy =2
0
for convenience). In other words
7rlog(U/Uo)
J - exp(O(u, y)) dy. (4.20)
V 27rkT <bguoy
0_
This integral can be approximated following Laplace's method, where in this case we
have a moving maximum
[49]. For fixed u the exponential is distributed around its peak
23
value y* = maxy exp(0(u, y; no)), so that we may make the following approximations
to leading order
ir log(u/uo)
f(y
1
YrF
-2U
P,(u I V < 0, Uo) ~
exp O(u, y*) +
,I
y
v'2-7ikT <(n)u
y*)2
-
2
dy
YO(u, y*) +
2
(4.21)
J
(u, y*; Uo)
0
exp
(Y -
2
*)T
YY)O(U, Y*) +..-) dy
(4.22)
-00
C
where
(4.23)
(U, y*; O)
a is a normalization
constant that is computed numerically. In order to arrive
at a closed analytic expression for y* = maxy exp(O(u, y)), we find that the following
expression is a good approximation for y* (see the note at the end of this section for
details):
/r
y*(U; Uo) =
2
1/4
-2
log(qU/O) 1/ 2,
m 2 > k2.
(4.24)
\4mT
Hence, we have the following expression that approximates the original integral:
1
P(u
IK <
0, 'O)
C
((u, y* uo),
m 2 > k2 .
(4.25)
In our case the assumption is that instabilities are sufficiently rare that the standard
deviation k of the Gaussian process K(t) will be smaller than its mean m, and the
condition m 2 >> k 2 in (4.25) will be satisfied. Indeed, the derived asymptotic expansion (4.25) and the exact integral (4.15) show excellent agreement, even for small
values of u (see Figure 4-2).
Note
Here we provide some details regarding the asymptotic expansion. According
to Laplace's method, this integral will be distributed near the peak value of the term
being exponentiated:
(y2 +2 rn) 2
(XY;XO
2k2
(xy;o) -
24
_
__
4T 2y2
log(X/o)
2
.
(4.26)
Exact
100
-- Asymptotic
10
102
103
)
2
4
6
U
10
8
Figure 4-2: Exact integral in equation (4.15) compared to the asymptotic expansion
given in equation (4.25), for a fixed no andm = 5.00, k = 2.00.
To find the maxima y* = max, exp(9(x, y; x 0 )), we differentiate (4.26) with respect
to y:
y2
--- 2y -+
+
0
k2
-+
(4.27)
2Tr log(x/xo) 2
.
ay
2T 2y3
To arrive at a closed form analytic approximation for y*, we must determine the relative
order of magnitude of y. To this end, investigating the terms in the integral in (4.13),
we note that the maximum ymx of PA(y) will determine the order of magnitude of y,
d
-P(y)
2w exp
(y 2 + n)2
dy k4b(rn)v/-2-
-
dy
2k2
0
)m)
-> 2y4 + 2mny2 - k 2 = 0;
thus if m 2 > k 2 , to leading order yka
(4.28)
~ k 2 /2m so that y 2 ~ O(k 2 /"n). Using this
result, we may approximate (4.26) by
2T2
log(XXO)
z-
2
y*(X; XO) = ( rk2
4rk-2
25
1/4
log(x/xo)1/
2
,
k2
)
y
M2
>> k2 . (4.29)
4.4
Envelope probability distribution in the stable
regime
In the derivation of the probability density in the unstable regime we conditioned
the result on the initial value of the position's envelope right before an instability
occurs. The stable regime's envelope is defined as the locus of local maxima of the
stationary Gaussian process that governs the system before an instability. For a
stationary Gaussian process of zero mean the joint probability density of the envelope
and the envelope velocity is given by [50]
P(-it) =
(liu
a
exp 2
qAo 2wA2
2
it 2~
_+
qA 2
Ao
,
(4.30)
where A, is the nth order spectral moment for the one-sided power spectral density
of x, in the stable regime and q 2 =1 - A2/AOA 2 describes the extent to which the
process is narrowbanded. In our case, we are interested in the pdf of the envelope for
the position P(uo); marginalizing out it from (4.30) gives
00
P(u, it) du
P(u)
exp(-u 2 /2Ao),
(4.31)
-00
which is a Rayleigh distribution with scale parameter v/\O. The zeroth-order spectral
moment is the variance of the Gaussian pdf in (4.9), i.e. A= o/(2cW2), and thus
2cw x
PU(x) =
2
exp(-cw x 2 /or ).
(4.32)
The pdf above determines the distribution of the initial point of the instability uo.
We use this result to marginalize out uO from the pdf of the unstable distribution
derived in Section 4.3, which was conditioned on a fixed initial point of an instability.
26
.
..
............
.....................................
Using (4.17) and (4.25) this gives the final result for the density in the unstable regime:
(X I K < 0)
1 P(xl (I <0, uo)P 0 (uo) duo
=
(4.33)
2
J((Ix1, y*(IxI;uo); uo)Pu(uo) duo,
=
(4.34)
0
In other words,
Px1 (x I unstable regime)
-
J
, y*(|xI; 'o); uo)Ps0 C(o) duo.
(4.35)
0
4.5
Probability distribution for the velocity
The probability density for the system's velocity x 2 in the stable regime can be found
by marginalizing the solution obtained from the Fokker-Planck equation (4.10). The
same result may be obtained by relying on the narrowband property of the stochastic
response in the stable regime. In particular, we may represent the stochastic process
for position x1 in terms of its spectral power density [25]
00
J= cos(Wt + (w)) 2Sx(w) dw,
X
(4.36)
0
where the last integral is defined in the mean square sense. Differentiating the above,
we obtain
00
w cos(wt + W(w) +
X2
2Sx(w) dw
(4.37)
0
00
cos(wt + y(w) + 2 V)2Sx(w) dw,
ws
(4.38)
0
using the narrowband property of the spectral density around W,. Note that the
integral in the last equation has the same distribution as x1 . Thus, it is easy to prove
that the probability density function of x 2 in the stable regime is given by the pdf of
WsX1.
27
In the unstable regime stationarity breaks down, and the envelope of the stochastic
processes describing position changes with time. This has an important influence on the
probabilistic structure for the velocity, which cannot be approximated as it was done
in the stable regime. The reason is that we have an additional time scale associated
with the variation of the envelope of the narrowband oscillation (see Figure 4-3). For
the purpose of computing the pdf for velocity in the unstable regime, we approximate
the envelope for position during the instability by a sine function with a half-period
equal to the average duration of the extreme events (4.16): Tilst
(1 + 2A,<o/c)Tco.
This gives the following approximate expression for the unstable regime:
'11=) sin(wat)
J
cos(wt + O(w))
(4.39)
2Sx(w) dw,
0
where up/uO is the ratio of the extreme event peak value for the envelope over the
initial value of the envelope (before the extreme event), and w,, =r/TiSt. Assuming
that up/uo >> 1 we have that the pdf for velocity during the unstable regime is also
given in terms of x 1 but with the different scaling factor
X2
+(
S+ - -X1.
2(
1 (4.40)
(1 + 2A.,scO/c)T)
The last relation gives a direct expression for the pdf of velocity in the unstable regime.
Using this scaling factorwi,,t --- (w, + wu)/2 for the conditionally unstable pdf and
the correct scaling for the stable regime w., we can derive an approximation for the
pdf of velocity (given in Section 4.6).
5- (t;O)
0 -----5 -0
50
100
t
150
200
Figure 4-3: For the computation of the velocity pdf, we approximate the envelope of
the process during the extreme event with a harmonic function with consistent period.
28
4.6
Summary of the analytical results
Combining the results of equations (4.9),
Probability distribution for position
(4.35), and (4.16) according to the law of total probability (2.6) gives the following
heavy-tailed, symmetric probability density function for the position:
P,, (x) =(i
exp
(1 + 2A,~o/c)+('q))
-
2x
-
2
1~
xCE R.
(4.41)
x2 dxo.
(4.42)
z)do
o;z)a
(xy(x
+ (1 + 2Xc/)()
0
Writing each term explicitly, we have
- (y*(Ix;x
x exp
j
(I ) f
VIcw 2
2
2i6or kT
o) 2 +-M)
2
2k2
Probability distribution for velocity
_
log(XI/XO)
o
(|x|/XO)y* (|~X O)
/
+ (1 + 2XA<o/c)I(q)
x2
exp
P, (x) = (1 - (1+ 2Ax,<O/c)())
log(|IX/xo)
2
_.
4T2y*(Ix;XO) 2
cw! 2
02
To obtain the pdf for velocity we scale the
probability distribution function for the position in (4.41). As noted in Section 4.5,
the correct scaling for the first term in (2.6), which represents the stable regime, is
given by w,, and for the second term, for the unstable regime, the scaling is given by
Winst.
Applying the random variable transformation P, (x) = P, (x/w)/w with the
appropriate scaling for each of these two terms yields
P 2 (x) =(i
-
(1 + 2AX
IXI/win~t
/c)i()
C exp(-
+ (1 + 2A,.<o/c)()
1
winst x )o ) no (xo
x wist
X
f
x)
2winstc
29
x,
xER. (4.43)
For the integral in the second term in (4.43), making the substitution xo
winst',
we have
w2
PX2(X IK< 0)=2u~T24
2
2
log(xf/xo)
(Ixl/xo)y*(IxI; Xo)
s
k0)20
x exp((Y*(IXI;
J
lxi
i7c
2
7 log( XIXO) 22 CW2 2
o2
+)2
2k2
4T y*(IxI; XO)
dx0. (4.44)
w;n2
Therefore, rewriting (4.43), gives the following pdf for velocity:
P2 (x)
C exp -C
(1 - (1 + 2AK<o/c)4b('r)
x2
lxi
+ (1 + 2Aso/c)<(r)
2
-
(xl, y*(xI; xo); xo)P,0 (xo/wi.t) dxo,
(4.45)
where the integral in the second term is explicitly given in (4.44).
4.7
Discussion and comparison with numerical simulations
Here we discuss the analytic results for the system response pdf given in Section 4.6,
present comparisons with Monte Carlo simulations, and describe the effect of varying
system parameters on the analytical approximations. In addition, we examine the
region of validity of our results with respect to these parameters.
First, in Figure 4-4 a sample system response is shown alongside the corresponding
signal of the parametric excitation. This sample realization is characteristic of the
typical system response for the case under study (rare instabilities of finite duration).
During the brief period when K(t) < 0, at approximately t = 85, the system experiences
exponential growth and when the parametric excitation process V.(t) returns above
the zero level, friction damps the system response from growing indefinitely, returning
the system to the stable state after a short decay phase.
30
-position-velocity
5-
0
-5
0t
40
60
80
60
80
t
100
12
1401TT-
100
120
140
10
5-
0 ----------------------40
60
----------------- ~~--- -- ~-100
80
120
140
Figure 4-4: Sample realization (top) and the corresponding parametric excitation
process (bottom) demonstrating the typical system response during an intermittent
event for position (in thick blue) and velocity (in thin red) for rj = -2.27 (k = 2.2,
m - 5.0), c = 0.53.
In Figure 4-5, a typical decomposition of the stable and unstable parts of the full
response pdf is presented. We emphasize that the tails in the system response pdf
for
are completely described by the term in the total probability law that accounts
in the
system instabilities, whereas the inner core of the pdf is mainly due to the term
total probability decomposition that corresponds to the solution in the stable regime.
Despite the fact that the system spends practically all of its time in a stable state,
the stable part in the expression for the response pdf contributes essentially nothing
to the tails of the distribution. The tails, in other words, completely account for the
statistics of system instabilities, which are highly non-Gaussian and are described by
a fat-tailed distribution.
The analytical results for the system response pdf given in (4.41) and (4.45) have
been derived under a careful set of assumptions. We observe that if the frequency
of instabilities is large or if damping is weak (i.e., extreme events of long duration),
pdf
or if there is any parameter combination such that the conditionally unstable
in the total probability decomposition (2.6) contributes significantly to the stable
state, then our analytical results will likely overestimate the probability near the
31
-Full
solution
-- Stable part
Unstable part
100
10
10
10-4
-6
-4
0
x
-2
2
4
6
Figure 4-5: Typical superposition of the stable (in dashed blue) and unstable (in dotted
red) part of the total probability law decomposition (2.6) of the full analytical pdf.
Gaussian core and underestimate the tail probabilities. This behavior is expected,
since our assumptions are no longer strictly valid. If this is the case, a cusp near the
mean state may be observed in our analytical expressions for the pdf, which is due to
the approximation made in Section 4.3, where we derive the conditionally unstable
pdf for the position variable in terms of its envelope (4.17) (see Figure 4-6). This
phenomenon is more pronounced in the pdf for the velocity x 2 , since the unstable part
of the decomposition (2.6) is scaled by a frequency that is smaller than the stable
part, which pushes more probability toward the Gaussian core. A minor contributing
factor to the severity of this cusp is due to the integral asymptotic expansion made in
Section 4.3.1; the asymptotic expression in (4.25) displays sharper curvature near uo
when compared to the exact integral in (4.25), resulting in a slightly more pronounced
cusp.
4.7.1
Comparison with Monte Carlo simulations
Here we compare the analytic approximations in (4.41) and (4.45) with Monte Carlo
simulations, when the stochastic parametric excitation
32
(t), with mean m and standard
*~0
10
10
-
- Full solution
Stable part
--
. Unstable part
Unstable part
10101
102
-8
-6
4
2
0
2
4
-2
8
6
-1.5
-1
-0.5
0
0.5
1
1.5
2
Figure 4-6: Demonstration of the cusp phenomenon, which is prominent in regimes with
very frequent instabilities and/or instabilities of long duration, due to approximations
made in the unstable regime.
deviation k, is given by an exponential squared correlation function of the form
R(T)= exp(-T 2
/2).
We solve (4.3) using the Euler-Mayaruma method [51], from to
-
0 to tf = 200
with step size At = 2-. Realizations of the Gaussian process K(t) are generated
according to an exact time domain method
following values for the system parameters: a.
[52]. To preform simulations we fix the
= 0.75, m
5.00. For the results in this
subsection, we furthermore fix the value of friction at c = 0.38, which corresponds to an
average of a single oscillation during the decay phase of an instability. Moreover, for this
subsection, samples after 20 computational steps are stored and used in calculations,
and 5000 realizations are used for k
=
1.8 and 2500 realizations otherwise. In Figs. 4-
7, 4-8, and 4-9 Monte Carlo results for the pdf for position and velocity are compared
with the analytic results given in (4.41) and (4.45) for various k values, which controls
the mean number of instabilities for fixed m through the parameter rY. Observe that the
analytical pdf for position and velocity match the results from Monte Carlo simulations
for both extreme events and the main stable response, capturing the overall shape and
probability values accurately (the figures for position are plotted out to 50 standard
deviations of the stable Gaussian core).
33
-- Monte Carlo
Carlo.
100 -Monte
-Analytic
100
-- Analytic
10-2
10
-
10-
-
10
10
-15
-5
-10
0
X
-30
15
10
5
-20
-10
30
20
10
X0
Figure 4-7: Analytic pdfs (4.41), (4.45) compared with results from Monte Carlo
simulations for position (left) and velocity (right), for parameters: k = 1.8, m = 5.0,
c = 0.38, o-x = 0.75.
-- Monte Carlo
-Analytic
-- Monte Carlo
-
10
-Analytic
10-2
102
10-4
10
10
-4
20
10-6
10-61
-5
-10
0
5
-20
10
-10
0
X
10
20
Figure 4-8: Analytic pdfs (4.41), (4.45) compared with results from Monte Carlo
simulations for position (left) and velocity (right), for parameters: k = 2.2, m = 5.0,
c = 0.53, a-= 0.75.
4.7.2
Parameters and region of validity
Here we quantify the effect of varying system parameters on our analytical approximations. In particular, we consider the effect of both friction c and the frequency of
instabilities, controlled by rj, on our results by computing the Kullback-Leibler (KL)
divergence
P(Pex, P) =
J
Pex(X) In PexW dx,
,P(x)
-00
(4.46)
which measures the relative information lost by using our approximate analytical pdf
P instead of the "true" or exact probability distribution Pex of the system from Monte
Carlo simulations.
34
-- Monte Carlo
-Analytic
-- Monte Carlo
10
-
Analytic
10
10
10 2
10 -10
100
110
100-4
10
-10
-5
-20
10
5
0
0
-10
10
20
X
X
Figure 4-9: Analytic pdfs (4.41), (4.45) compared with results from Monte Carlo
simulations for position (left) and velocity (right), for parameters: k = 2.6, m = 5.0,
c = 0.69, ox = 0.75.
For a more physical interpretation of the effect of the friction parameter, we instead
consider the typical number of oscillations the system undergoes during the decay
phase of an instability, i.e., the number of oscillations before the system returns back to
the stable regime. To determine the value of damping in terms of the average number
of oscillations N during the decay phase, consider that in Section 4.3 we derived the
approximation (4.16): Tims = (1 + 2A,<o/c)T,<o. In other words we have that the
duration of the decay phase is given by Tdecay - (2AK<o/c)TK,<o. Thus, the average
length of the decay phase is Tdeay = (2,K<o/c)T (where T denotes the conditional
average Tco), and the corresponding value of damping that will give on average N
oscillations during the decay phase is therefore
W~dc
27r
-s
2An<oW
27r
WsA<oc
irN
where T is given in (3.5) and w, in (4.7).
In Figure 4-10, we compute the KL divergence for a range of k and N parameter
values. For the simulations we fix m = 5.0, store values after 40 computational steps,
and use 4000 realizations for k < 2.0 and 2000 realizations, otherwise. In addition, we
compute the divergence for every half oscillation number from N = 0.5 to N = 3.0
and every k value from k = 1.4 to k = 3.6, in increments of 0.2, and interpolate values
in between. Overall, we have good agreement for a wide range of parameters, and even
when the analytical results do not capture the exact probability values accurately, we
35
have qualitative agreement with the overall shape of the pdf. The analytical results
deviate when we are in regimes with very frequent instabilities and instabilities of
long duration, i.e., small friction values or large N values. This is expected; in such
regimes the instabilities are no longer statistically independent, which we assume in
Al. Furthermore, an average of N = 1 oscillation during the decay phase shows good
agreement with Monte Carlo results for a large range values of the parameter 'q.
We also observe that deviations of the analytical approximation from the Monte
Carlo simulation become more prominent as k decreases. Note that for small k sampling
the tails is very hard since the critical events occur very infrequently in this case. To
understand the deviation of the analytical solution we recall that for this parametric
regime the unstable region is characterized by rare events which are shorter and weaker
and therefore harder to distinguish from the regular events. This explains the good
analytical approximation properties as k increases (but not too large), where we have
clear separation of the system behavior between the stable and the unstable regimes.
Finally, we emphasize that the KL divergence may be (relatively) large despite the
analytic pdf capturing the trend in the tails accurately; this is due to the fact that tail
events are associated with extremely small probability values, which the KL divergence
does not emphasize over the stable regime, where the probabilities are large and thus
any deviations (e.g., due to a cusp) can heavily impact divergence values.
36
KL divergence
log 10
KL divergence
- position
log10
- velocity
-1.8
-1.5
-1.9
2.5P
2.5
-2
-2
-2.1
2
2
-2.2
-2.3
1.5
0
-T
-2.5
1.5
-2.4
1
--3
-2.6
0.5
152
2.5
k
3
0.5
35
1.5
2
2.
k
3
.
-2.5
Figure 4-10: KL divergence for position (left) and velocity (right) between analytic and
Monte Carlo results for various values of damping (in terms of the number of oscillations
N after an instability) and k, which controls the number of instabilities for a fixed m.
For fixed m = 5.0.
3'7
38
Chapter 5
Application to intermittently
unstable turbulent modes
In this chapter we discuss another application and apply the method formulated in
Chapter 2. We consider a complex scalar Langevin equation that models a single
mode in a turbulent signal, where multiplicative stochastic damping 'Y(t) and colored
additive noise b(t) replace interactions between various modes. The nonlinear system
is given by
)
du(t
dt
dt
(--y(t) + iw)u(t) + b(t) + f(t) + uW(t),
(5.1)
where u(t) E C physically describes a resolved mode in a turbulent signal and f(t) is
a prescribed deterministic forcing. The process 'y(t) models intermittency due to the
(hidden) nonlinear interactions between u(t) and other unobserved modes. In other
words, intermittency in the variable u(t) is primarily due to the action of -y(t), with
u(t) switching between stable and unstable regimes when -Y(t) switches signs between
positive and negative values.
The nonlinear system (5.1) was introduced for filtering of multiscale turbulent
signals with hidden instabilities [53, 54] and has been used extensively for filtering,
prediction, and calibration of climatological systems [13, 55, 10, 11, 9, 56]. This
system features rich dynamics that closely mimic turbulent signals in various regimes
of the turbulent spectrum. In particular, there are three physically relevant dynamical
39
regimes of (5.1); the three regimes are described by
[55]
(and are reproduced here for
completeness):
R1 This is a regime where the dynamics of u(t) are dominated by frequent, shortlasting transient instabilities, which is characteristic of the dynamics in the
turbulent energy transfer range.
R2 In this regime the dynamics of u(t) are characterized by large-amplitude intermittent instabilities followed by a relaxation phase. Here the dynamics are
characteristic of turbulent modes in the dissipative range.
R3 This regime is characterized by dynamics of u(t) where transient instabilities
are very rare and fluctuations in u(t) rapidly decorrelate. This type of dynamics
is characteristic of the laminar modes in the turbulent spectrum.
We apply the method described in Chapter 2 to approximate the probability
distribution for the dynamics of u(t) in the special case with no additive noise, i.e.
b = 0, and no external forcing
f
= 0. This is the simplest case that incorporates
intermittency driven by parametric excitation process -y(t) (the additive noise term
b(t) only impacts the strength of intermittent events). The system we consider is given
by
)
du(t
dt=
dt
(-(t)
+ iW)u(t) + uw(t),
(5.2)
where intermittent events are triggered when -y(t) < 0. The parametric excitation
2
process -y(t) will be characterized by a mean in and variance k , so that r
-m/k
represents the extreme event level, and also a time correlation length scale L. This
application is a good test case of our method for the derivation of analytic approximations for the pdf of intermittent systems, since the dynamics of u(t) are such that
it oscillates at a fixed frequency w, and therefore no approximations need to be made
as in Chapter 4 for the growth exponent in the unstable state. However, the Langevin
equation has the difficulty that the nature of its interactions with the parametric
excitation process introduces "instabilities" of such small magnitude that they are
indistinguishable from the typical stable response, and so a clean separation must be
40
enforced in the analysis of the unstable state. We present these differences and derive
the system response pdf -for the real/imaginary parts of the response u = u, + i't,
which statistically converge to the same distribution so that P(ur) and P(ui) are
equivalent. We compare the analytical result for all three regimes R1-R3, but we note
that the pdf in regime 3 is nearly Gaussian so the full application of our method is
excessive.
5.1
Probability distribution in the stable regime
Here we derive an approximation of the pdf for u(t) given that we are in the stable
regime and, moreover, that we have reached a statistical steady state. In the stable
regime, following Section 2.2, we replace ,(t) by the conditional average
m --
Tn
<n
011)(5.3)
+
Thus the governing equation in the stable regime becomes
du
(7K>o + iwt) t) +
7W(t),
(5.4)
dt
with the exact solution
u(t) = u(O)e(-lY>o+iW)(t-to) +
J
(5.5)
e(or >o+iw)(ts) dW(s).
to
Since this is a Gaussian system, we can fully describe the pdf for u(t) in this regime
by its stationary mean u(t) = 0 and stationary variance
2
Var(u(t))
= Var(uo)e~27>o(t-to) +
(1 -
27a
-27l>o(t-to))
-+
c(>O
-
271|y>o
41
as t -+ oo. (5.6)
Therefore, we have the following pdf for the real part of u(t) in the stable regime:
5.2
7ir
2
2
exp
X2
(5.7)
.
Pu,(x I stable regime) =
Probability distribution in the unstable regime
In the unstable regime, as in the previous application, we describe the probability
2
distribution in terms of the envelope process. Following assumption A2, we ignore or
in (5.2), which does not have a large probabilistic impact on the instability strength,
so that the envelope of u (t) is given by
-
2Re
- -2y(t)lul
[u*
[dt
2
(5.8)
,
dtu|2
dt
du(t)
dt
where LI* is the complex conjugate. Substituting the representation Jul ~ uoeAT
into (5.8), we get A = -- y. Now since ?(A) = P(-
PA(A)
1 0 (jA +m)
k
k4D (q)
1 P (-A)
-
y7 < 0), we have
<}(Y)
(5.9)
A > 0.
The conditionally unstable pdf will be exactly as in (4.13), with the minor modification
that the distribution of the growth exponent is given instead by (5.9). Making this
modification, we have the following pdf for the envelope in the unstable regime:
2kT
<DIg)u
f
'
1
7r log(u/uo)
PIUO I -Y< 0, Uol) =
y(
(
x exp
Y +rm
k)
2
4T y2
log(U/o)2) dy,
u > U0 ,
(5.10)
where T is given in (3.5). Moreover, using (2.16), we have that the average length of
an instability is given by
Tdecay
T,<o
_
k (7)
_ <(77)
inst =
m- + k,_4
42
(1 + P)T<O.
(5.11)
Finally, to construct the full distribution for the envelope of u(t) in the unstable
regime, we need to incorporate the distribution of the initial point of the instability,
which is described by the envelope of u(t) in the stable regime; this gives,
(5.12)
POUI I- < 0) = P(IuI I Y < 0, IuoD)P(uoI).
The pdf for the envelope of a general Gaussian process was described in Section 4.4.
However, as we noted in the introduction, the complex scalar Langevin equation
has the property that its interaction with the parametric excitation gives rise to
"instabilities" of very small intensity which are indistinguishable from the typical
stable state response. To enforce the separation of the unstable response from the
stable state requires us to introduce the following correction to the initial point of
the instability iuol = Itol + c, where 1-hol is the pdf of the envelope of the stable
response (4.31) and c is a constant that enforces the separation. We find that choosing
c such that it is one standard deviation of the typical stable response (5.6) is sufficient
to enforce this separation and works well. In addition, this choice is associated with
robust performance over different parametric regimes. Therefore, we have that the
distribution of the initial point of an instability is given by
2
Pi.O1 (x) - 471,Y> (x - c) exp
yj (x - c)2)
x
> c.
(5.13)
As in the application to the parametric oscillator, we note that the oscillatory
character during an instability has to be taken into account. However, to avoid the
additional integral that would result should this be taken into account by an additional
random variable transformation, we instead utilize the same approximation as in
Section 4.3. Therefore, we have that the unstable regime pdf is given by
P,.(x
I unstable
regime) = P,(x
I
< 0) =Pii(Ix I -Y < 0).
(5.14)
We show in the following sections that this approximation compares favorably with
direct numerical simulations.
43
.
.......
5.3
Summary of the analytical results
Combining (5.7), (5.10), (5.13), and (5.11) into the total probability law decomposition (2.6) and utilizing the approximation (5.14) gives the following heavy-tailed,
symmetric pdf for the complex mode u(t) = ur(t) + ini(t), for x c R:
(1
Pur(X)F-
(1
i(1))1
2
+ (1 +
x
exp
(
2k 2
2
( e>o2 2
2 7Lo
)D)
)
exp
0-12
|>o
2&2kT io
ff
log(Ixj/Uo)
y 2 (|x|/(uo -
c))
_ 2 log(Ix|/uo) 2 _ 2*>o ('O - c)2 dy duo,
4T y
(5.15)
where the integral is understood to be zero when x < c.
In this case, we do not use an asymptotic expansion for the integral in the unstable
regime due to the fact that the resulting expression does not agree as precisely as the
integral expansion for the parametrically excited oscillator application. However, the
resulting expansion does follow the same decay as the full integral; thus should an
asymptotic expansion for the integral be sought, the derivation follows the description
given in Section 4.3.1
5.4
Discussion and comparison with numerical simulations
Here we compare the analytic results (5.15) with direct Monte Carlo simulations in
the three regimes R1-R3 described in the introduction of the current Section. Monte
Carlo simulations are carried by numerically discretizing the governing system (5.2)
using the Euler-Mayaruma method with a time step At = 10-4. We store results
after 10 computational steps and use 500 ensembles of time length t = 700, discarding
the first t = 200 time data to ensure a statistical steady state for the calculation
of the response statistics. We prescribe the parametric excitation process to have
44
an exponential squared time correlation function, R(r) = exp(-T 2 /(2L)), with a
time length scale L, and simulate the Gaussian process directly from the correlation
function.
In Figure 5-1, the results for regime 1 are presented alongside a sample realization.
As previously mentioned, this regime is characterized by frequent short-lasting instabilities. We observe that the analytical results are able to capture the the heavy-tails
of the response probability distribution accurately, including the Gaussian core. In
contrast, in regime 2 (see Figure 5-2), we have large-amplitude instabilities that occur
less frequently. Again the analytical results in this regime are able to capture the
probability distribution of the response accurately. In both regime 1 and regime 2, the
pdf of the response is plotted out to 200 standard deviations of the stable state's standard deviation to show the remarkable agreement of the tails between the analytical
solution and Monte Carlo results. In Figure 5-3 we present the results for regime 3 for
completeness, even though, as previously mentioned, this regime is nearly Gaussian
and intermittent events occur very rarely, and under certain parameters almost surely
no instabilities will be observed.
10,
---Monte Carlo
- Analytic
-
4
100
-
2
Z
s
-210
10-
-4
-3
-1
-4
1-6 _______________________________
-5
0
50
100
150
200
t
250
300
350
-30
400
-20
-10
0
X
10
20
30
Figure 5-1: Regime 1: Sample path (left) and analytic pdf (5.15) compared with results
from Monte Carlo simulations (right), for parameters: w = 1.78, a = 0.5, m = 2.25,
k = 2.0, L = 0.125.
45
r
10
Monte Carlo
-Analytic
---
-
4
100
3
10
2
10-2
-
-2
10,
-
-3
-4
0
200
t
150
100
50
250
300
350
10
400
-10
10
5
0
-5
Figure 5-2: Regime 2: Sample path (left) and analytic pdf (5.15) compared with results
from Monte Carlo simulations (right), for parameters: w = 1.78, o- = 0.1, m = 0.55,
k = 0.50, L = 2.0.
10 2
Monte Carlo
-Analytic
--
-
1
0.8
100
0.6
0.4
0.2
10-2
-
-----0.4
-0.6
10-6
-0.8
-1
0
50
100
150
200
250
300
350
-0.4
400
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Figure 5-3: Regime 3: Sample path (left) and analytic pdf (5.15) compared with results
from Monte Carlo simulations (right), for parameters: w = 1.78, a = 0.25, m = 8.1,
k = 1.414, L = 4.0.
46
Chapter 6
Application to the stochastic
Mathieu equation
In this chapter we consider the problem of intermittent parametric resonance. Parametric resonance has been extensively studied using the classical Mathieu equation,
X + (6 + Ccos(Qt))x = 0,
(6.1)
because it represents the typical response of systems when excited by a periodic
force. Time periodic forcing is found in many physical systems, examples naturally
include those mentioned in Chapter 4. Additionally, the Mathieu equation (and the
more general Hill's equation) has been studied in connection with orbital problems in
astrophysics [57], wave propagation in periodic media [58], and many problems in fluid
dynamics [59, 60], not to mention ship roll motion in regular waves [44]. Nevertheless,
the problem of stochastic generalizations of the Mathieu equation remain less well
understood [61, 57], especially tail statistics and non-Gaussian responses. Here we
discuss a particular stochastic generalization, with time correlated random amplitude
forcing (i.e., c is a random process) and quantify the corresponding response pdf. We
analytically derive results for the response pdf when the system exhibits intermittent
parametric resonance. Quantifying the response pdf in this case is important for
47
............
"I'll",
........
.......
predicting rare event probabilities, since the corresponding large amplitude responses
are considered extreme events in natural systems.
For parametrically excited motions, parametric resonance can produce large responses even if the amplitude of the parametric excitation is small. For Mathieu's
equation (6.1), when the frequency of the excitation Q is near twice the natural frequency of the system wo
=
VS the response becomes unstable even for small c (primary
resonance). The Ince-Strutt diagram in Figure 6-2 shows resonance "tongues," where
particular combinations of the forcing amplitude and natural frequency produce unstable solutions. The basic mechanism of unstable growth by parametric resonance
is nicely explained in geometric terms in [62, Chapter 3.5.6]. The action of unstable
oscillatory growth via parametric resonance, we consider different than the purely
exponential unstable growth explored in Chapter 4; the differences, we illustrate in
Figure 6-1. Namely, the intermittent instability case refers to events with 6 < 0 and
exponentially growing solutions, where as intermittent resonance occurs when the
system randomly enters the unstable resonance tongues, shown in Figure 6-2, and
have solutions that grow oscillatorily with 6 > 0. The distinction is not too important,
however the structure and nature of the excitation and modeling the noise appropriately can require different analytical methods, which we illustrate in this chapter.
78
2
1.5
6
5
0.5
4
0
-4
3
-
-0.5
1
-1
-1.5
0
20
40
60
80
100
0
120
0
2
6
4
8
time
time
Figure 6-1: Oscillatory growth due to parametric resonance (left) for 6 > 0 and
exponential growth (right) when 6 < 0 of the Mathieu equation.
48
10
2
1.4
1.2
0.8
0.6
0.4
0
-0.5
0
0.5
1
1.5
2
2.5
3
6
Figure 6-2: Ince-Strutt diagram showing the classical resonance tongues (shaded) for
particular combinations of E and 6, for the Mathieu equation (6.1) non-dimensionalized
by Q. Transition frequencies at 6 = wo/Q2 = (n/2)2 . Stability boundaries are computed
via Hill's infinite deteriminant [1].
6.1
Problem definition and formulation
We study the following stochastic generalization of the Mathieu differential equation:
z(t) + 2E(owo (t) + W2(1 + 0 (t; 6) sin Qt)x(t) =
(t; 0),
(6.2)
where (0 is the damping ratio, wo is the undamped natural frequency of the system,
Q is the frequency of the harmonic excitation term and 0(t; 0) its random amplitude,
S(t; 0) a broadband weakly stationary random excitation. As mentioned previously,
the deterministic Mathieu equation
J(t) + 2(owo.(t) + w0(1 + A sin Qt)x = 0,
(6.3)
has unstable solutions depending upon the parametric excitation frequency and amplitude parameters . Near Q/2wo = 1/n, for positive integers n, we have regions of
instabilities, with the widest instability region being for n = 1. Damping has the effect
of raising the instability regions from the Q/2wo axis by 2(2() 1/n. Therefore, for ( < 1
49
the instability region near n = 1 is of greatest practical importance [39, 1]. In the
2wo. The
following we consider (6.2) tuned to the important resonant frequency Q
case, where the frequency is slightly detuned can be generalized following the same
approach, but for simplicity of the presentation we consider no detuning. In realistic
systems the parameter A in (6.3) that controls the stability of the system for a fixed Q
and (o may be a random quantity and not necessarily deterministic (e.g., in ship rolling
models [31] and orbital problems in astrophysics [57]). If this is the case, intermittent
resonance is a possibility since the stochastic process /(t) in (6.2) can randomly enter
the instability tongue, triggering a large amplitude response (see Figure 6-3). In other
words, we are interested in the case where
#(t)
on average lives in a "safe" regime,
but can randomly enter a critical regime, where an intermittently unstable response is
triggered. As we show, ignoring the random nature of the process /(t) would lead to
severely underestimated tail probabilities, since the corresponding averaged equation
have Gaussian statistics, whereas the actual response features heavy-tailed statistics.
response
0.2
0.1
0
-0.1
-0.2
0
800
600
1200
1000
16
1400
)0
time
0(t) excitation
0 .5
-
_-_-_- -_-_
- -_-__- _-_-_-_-_-_
-_-_-
0-
-1 [
400
-
I
600
~~ ~
~
-~- ~ ~__
~
~
~ ~~
~~
~
-
-0.5
800
1000
1200
1400
1600
time
Figure 6-3: Sample realization of the Mathieu equation (6.4) (top). The random
amplitude forcing term 0(t) (bottom) triggers intermittent resonance when it crosses
above or below the instability threshold (dashed lines).
50
Based on the discussion above, we consider the problem,
z(t) + 2(owoi(t) + w0(1 + 0(t; 0) sin 2wot)x(t) = (t; 0),
(6.4)
where we additionally assume that 0(t) is a correlated stationary Gaussian process.
To solve for the probability distribution for the response, a typical approach would be
to first derive a set of equations for the slowly varying variables and then to apply the
stochastic averaging procedure to arrive at a set of It6 stochastic differential equations
for the transformed coordinates. The Fokker-Plank equation can then be used to solve
for the response pdf [39, 63]. This standard approach, if applied to the problem (6.4),
leads to Gaussian statistics. However, in certain parametric regimes, randomness in
the amplitude 3(t) may lead to intermittent instabilities and, therefore, non-Gaussian
statistics, which the averaging procedure would fail to capture. To account for the
statistics due to intermittent events triggered by 13(t), we decompose the probabilistic
system response by conditioning on the stable regime and unstable events. We remark
that for the system (6.4) to feature intermittent instabilities, the correlation length
of the process 0(t) should be sufficiently large compared the time scale of damping
tdamping
6.1.1
~1/(o so that instabilities have time to develop.
Derivation of the slowly varying variables
To proceed we average the governing system (6.4) over the fast frequency wo. We
assume that correlation length of the process 13(t) varies slowly over the system's
natural period 2wr/wo so that
#(t)
can be treated constant over one period; this
assumption is naturally satisfied if the response features large-amplitude intermittent
instabilities.
We first introduce the coordinate transformation
x(t) = xI(t) cos(wot) + x 2 (t) sin(wot),
x(t) = -wox 1 (t) sin(wot) + wox 2 (t) cos(wot),
51
(6.5)
for the slowly varying variables x 1 and X2. Inserting (6.5) into (6.4) and using the
additional relation i1 cos(wot) + : 2 sin(wot) = 0, gives the following pair of differential
equations for the slow variables:
Xi(t)
-
[2(owo (x 1 sin 2 (wot)
-
X 2 (t)
wo/3
-
2
X 2 sin(2wot)
sin2 (wot)
2
-
Wof1
/
1
sin(2wt) -
-
[2(owo ( x sin(2wot) - X 2 cos2(wOt)
WO
(6.6)
sin(wot)(t),
2
wo3Xi cos 2 (wot) sin(2wt) + -
-
sin2 (2wt)
sin2 (2wt)
(6.7)
cos(wot)(t).
Averaging the deterministic terms in brackets over the fast frequency, "a f 2/j'w dt,
in (6.6) and (6.7) gives,
x1
-
(o
-
2-
O9wox1 -
- sin(wot) (t),
(6.8)
cos(wot)(t).
(6.9)
4
WO
4
WO
The equations above for x1 and X 2 provide good pathwise and statistical agreement
with (6.4). Applying the stochastic averaging procedure to the random forcing gives
the following set of Ito stochastic differential equations for the slow variables
#()w
0x 1
+ 4(t)
wo/2
1-
2
with K
-(
2K7rK IV 1 (t),
(6.10)
2rK172 (t),
(6.11)
S (wo)/2wo, where S (wo) is the spectral density of the additive excitation
S(t) at frequency wo, and Wi 1 and
intensity
+
2
are independent white noise processes of unit
[39]. The slowly varying variables, after averaging the forcing term, transform
to two decoupled Ornstein-Uhlenbeck (OU) processes. The equations (6.10) and (6.11)
are a good statistical approximation (but provide poor pathwise agreement) to the
original dynamical system (6.4). We emphasize that we have not averaged the stochas-
52
tic process ,3(t) in any of this analysis, since it is randomness in the amplitude that
is triggering transient instabilities; the random nature of 13(t) must be accounted for
in order to capture the system's heavy-tailed probabilistic response.
6.2
Probability distribution for the slow variables
The equations derived for the slow variables in Section 6.1.1 fully describe the system's probabilistic response. Therefore, we concentrate on deriving the pdf for the
slow variables X 1 , x 2 starting from (6.10)and (6.11) by decomposing the probabilistic
response on the stable regime and the unstable state according to (2.6). Now if /(t) is
a zero mean process both x 1 and x 2 will follow the same distribution. Incorporating
bias in the amplitude excitation term 0(t) is straightforward. However, we assume
/(t) = 0, and write /3(t) = k(t), where '(t) is a Gaussian process with zero mean and
unit variance, and k 2 is the variance of 0(t). We consider the following OU process,
,
which models either xi or x2
i~ ~ =O
(6.12)
X-#()s +a- VV(t),
in other words
-
where -y(t)
f - AI'(t) is a Gaussian process with mean ri = (owo and standard
I
kwo/4. From (6.13), it is clear that instabilities are triggered when
deviation
'(t) < 0.
We define the extreme event threshold, as before, by i= fi-/ik. According Section 2.4, using (2.15), we have that the typical ratio between the growth and decay
phase is given by
Tdecay
_
-
k
(6.14)
__
T_, 7,:,
M
The total duration of an unstable event is the sum of the duration of the growth
phase T)<o and the decay phase Tdecay, i.e. Tinst =(1 + /-t)Tx<o. Therefore, we have the
53
...... ..............
- -- .. . I
-
.............
following result for the probabilities of the unstable and stable regimes, respectively:
P(unstable regime) = (1 + [)P(- < 0) = (1 + p)(I - <b(rj)),
P(stable regime) = 1 - (1 + p)P(y < 0) = 1 - (1 + p)(l - ob(i7)).
6.2.1
(6.15)
(6.16)
Stable regime distribution
In the stable regime, by definition, we have no intermittent events. We again approximate the dynamics in this regime by replacing T(t) with its conditionally positive
average 7>0:
T o
(6.17)
?in +
Making this approximation we have the following equation that describes the dynamics
in the stable regime:
=-
o x + a-
(6.18)
The corresponding equilibrium pdf for (6.18) is a Gaussian distribution (by using
the Fokker-Planck equation 124]) and thus the pdf that describes the response in the
stable regime is,
Px(x I stable regime) =Up
6.2.2
0expx 2
(6.19)
Unstable regime distribution
Next we derive the distribution for the system's response in the unstable regime.
Following A2, we assume that the parametric excitation 7(t) is the primary mechanism
driving instabilities and ignore the additive forcing term, which has minimal impact
on the pdf of the response in this regime. We characterize the growth phase by the
envelope of the response u ~ uoeAT. By substituting this representation into (6.13),
we get A = -y, so that P(A) = P(-j I -y < 0). Therefore, the distribution of the
54
Lyapunov exponent is given by,
PA (A) =
(I -
<b(rI))k
(6.20)
A > 0.
+-( n,
I
The conditionally unstable pdf for u is the same as in (4.13), with the modification
that the distribution of the growth exponent is instead given by (6.20):
f!,1
r log(u/uo)
2kT2 (1 -
<1(r))u ?
y
x exp
-
y+m
k
_2
log(U/uo) 2
dy,
u > uO.
(6.21)
\ 4T 2y2
The random variable uO describes the initial condition right before an instability
occurs. The pdf for uO is thus given by the envelope in the stable regime (see Section 4.4). However, as in Section 5.2, since the dynamics in this case are essentially
governed by the same equation, we note that the OU process (6.13) has the pathology
that its interactions with the parametric excitation give rise to "instabilities" that are
indistinguishable from the typical stable state response. This requires us to enforce the
separation of the stable and unstable regimes by introducing the correction no
=
'Tt0 + x,
where ho is the pdf of the envelope in the stable regime and c is a constant equal to
one standard deviation of the stable response distribution, i.e., c = o.2/(27I,>o). From
a simple random variable transformation, we have the following result for the pdf for
aO:
P 0 (x) = 2 @,>0 (x - c) exp
- K
(x
-
c)2),
X >
c.
Using (6.22) we can marginalize out uO in (6.20) by P(u I - < 0) = P(n
(6.22)
'
<
0, uo)P(no). In the last step, we transform the envelope representation of the dynamics in the unstable regime back to the variable we are interested in by a similar
narrowbanded argument as in the previous applications, giving
P,(x I unstable regime)
=
Px(x -y < 0)
55
=
1
2
-PU(IxI I < 0),
(6.23)
from which the final result for the pdf in the unstable regime can be easily obtained.
6.3
Summary of the analytical results
Using the results from Section 6.2, constructing the full probability distribution for
the slow variable is straightforward and requires combining the pdf for the stable
regime (6.19) and the unstable regime (6.23), with the appropriate weights (6.15),
using the total probability law (2.6). Since we assumed that the noise term 13(t) is
unbiased, i.e. a zero mean process, the corresponding distribution for the response
of the Mathieu equation (6.4) will be given by the distribution of the average of the
slow variable x1 and x 2 (which are equivalent), since the response is a narrowbanded
process, according to x(t) = xI(t) cos(wot) + x 2 (t) sin(wot). Thus we have that pdf of
the response for the Mathieu equation (6.4) is given by
PX(x)
(I - (1 +
20)(1
exp(
-
2
+ (1+ A) V7F21>o
4a-kT>
x
ex-p
-
2k2
--
-
4T2 y2
l
[I
I
0(2))
2
log(Ixl/uo)
- 2 (Ix|/(Uo
y
-
c))
C)2
O)2
dy duo,
(6.24)
C
where the integral is zero for x < c.
6.4
Discussion and comparison with numerical simulations
Next we compare the derived analytical results (6.24) with Monte-Carlo simulations
of the original system (6.4). To perform comparisons we use white noise W(t) with
intensity 3 for the additive forcing term (t) and furthermore non-dimensionalize time
56
by wo in (6.4) so that,
z(t)
+ 2(o b(t) + (1 + /3(t) sin 2t)x(t) = 6W(t).
(6.25)
We solve the equation above (6.25) for 2500 realization using forward-Euler integration
with a time step dt = 5 x 10-3 from t = 0 to t = 3500 and discard the first 500 time
units, to ensure a statistical steady state. Moreover, we simulate the stochastic process
13(t) according to the method presented in Appendix A.
For comparison we show six cases of varying intermittency levels in Figure 6-4. For
the comparisons, we set the damping at (0 = 0.1, 6 = 0.002, the correlation length
of 0(t) to be 50 times the time scale of damping Lcorr = 500, and show six cases
where we vary the intermittency level through the variance of the random process
0(t). For the most intermittent regime we set the standard deviation of 0(t) = kj(t)
to k = 0.40 so that rare event crossings occur with a 16% chance and for the least
intermittent regime k = 0.178 with a 2.3% rare event crossing frequency. Overall we
have good agreement between the analytic results and Monte-Carlo simulations for
all of the presented cases, which are robust across a range of parameters that satisfy
the assumptions.
57
Monte Carlo
Analytic
-
-
102
100
L.
Monte Carlo
-
Analytic
102 ---
.
LL100
10-2 f
10-2
0
-01
-00
-0.1
-0.05
01
00-.1
10-LL
-41L
-0.15
0.15
0.1
0.05
0
-0.15
-0.1
-0.05
0
x
x
(a) k
=
(b) k
0.20
=
0.05
0.1
0.15
0.22
M
103
-
--
Monte Carlo
-
Monte Carlo
Analytic
102
102
-nlyi
101
101
Li.
o 100
0 100
10.1
10.2
10.2
1031
-0. 15
-0.1
(c) k
=
1
0.15
0.1
0.05
0
x
-0.05
-0.1
-0.05
0
0.05
0.1
x
(d) k = 0.29
0.25
'U
10,
-
-
10
Monte Carlo
Analytic
-
Monte Carlo
Analytic
102
102
101
101
L.
U0
0
D_
a_
100
100
10'
10-1
10.21
-0.1
-0.05
0
0.05
IV
0.1
.1
-4
E
-0.1
-0.05
0
0.05
0.1
x
(f) k = 0.40
(e) k = 0.33
Figure 6-4: Comparison of Monte-Carlo results (6.25) (red curve) and analytic results (6.24) of the Mathieu equation for various intermittency levels with 6-4a being
least intermittent and 6-4f most intermittent.
58
Chapter 7
Conclusions and future work
We have formulated a method to analytically approximate the pdf of intermittently
unstable dynamical systems excited by correlated stochastic noise. The conditions and
assumptions we have used to derive the method presented in this thesis are applicable
to a broad class of problems in science and engineering. The core idea relies on
conditioning the dynamics on stable regimes and unstable events according to a total
probability law argument. The full system response is then derived by combining the
results of the analysis from the two regimes. We demonstrate how this decomposition
into a statistically stationary part (the stable regime) and an essentially transient
part (the unstable regime) can be performed in an analytical framework. This total
probability law decomposition is able to accurately capture the heavy-tailed statistics
that arise due to the presence of intermittent instabilities, and we show how it provides
a direct link between the character of the intermittent instabilities and the form of
the heavy-tailed statistics of the response.
We applied the formulated method to three prototype intermittently unstable
systems: (1) a parametrically excited mechanical oscillator, (2) a complex mode that
represents an intermittent mode in a turbulent system, and (3) the stochastic Mathieu
equation. In all three cases, we have made appropriate assumptions to arrive at
analytical approximations for the response pdf. We show that the derived analytical
results for all three prototype systems compare favorably with Monte Carlo simulations
59
under a broad range of parametric regimes, and demonstrate that the tails of the
analytical results capture the trend (and probabilities) with the numerical simulations.
In the parametric oscillator application, our analysis has furthermore unveiled
the important fact that intermittency in the velocity variable does not obey the
familiar scaling of the position variable with the stable state frequency. Instead, he
conditionally unstable pdf must be scaled by a frequency that accounts for both the
stable state frequency (fast) and a (slow) frequency associated with intermittent events.
We have made appropriate approximations to account for this observation. In the
complex mode application and the stochastic Mathieu equation, we observed that the
nature of the Langevin equation (with no inertial terms) leads to interactions with
the parametric excitation that give rise to instabilities that are indistinguishable from
the typical stable state response; again, we have made appropriate modifications to
account for this "mixing" between the two regimes for these types of systems.
Future work will include the extension and generalization of the ideas developed
in this thesis into a unified framework that can be used to study extreme events for
general dynamical systems. In particular, effort will be aimed at applying the idea
of a probabilistic decomposition to the study of extremes to more complex systems,
including nonlinear water waves, multi degree-of-freedom vibrations, and also to the
motion of finite-sized particles in fluid flows.
60
Appendix A
Gaussian Process Simulation
Here we describe exact methods for the numerical simulation of Gaussian processes.
In particular, we provide details on how to simulate realizations of Gaussian processes
on R, i.e. one-dimensional Euclidean space. Further details regarding the simulation
procedures described here and additional methods can be found in [64].
To generate a realization of a Gaussian process with expectation A(t) and covariance R(s, t) at times t. ..
, t.
we can sample a multivariate normal random vector with
the same mean and covariance, such that X = (X 1 ,..., X,)
= (X(ti
. . . ,X(t))
This leads to the following simple algorithm.
Algorithm A.1 (Gaussian Process Simulation)
1. Construct the mean vector p and covariance matrix E by settings pi = p(ti),
for i = 1, ... ,n and Ej -=R(tit ), forij 1,...,n.
2. Find the (lower) Cholesky factorization A of the covariance matrix E = AAT.
3. Generate Z 1 , . . . , Z, ~ N(0, 1), and let Z
(Z 1 ,... , ZJ)T.
4. Output X = t + AZ
It is always possible to find a Cholesky factorization of the positive definite covariance matrix E, however the issue with the algorithm A.1 is that a Cholesky
decomposition of an n x n matrix takes 0(n 3 ) floating point operations. Simulating
61
a long realizations quickly becomes time-consuming and introduces large memory
constraints. However, for stationary Gaussian processes we can take advantage of
the FFT to perform a fast O(nlognr) exact simulation, according to the circulant
embedding method originally proposed in [65].
The main idea is based on the fact that the covariance matrix E for a stationary
Gaussian process X(t), which is a symmetric Teoplitz matrix, can be embedded in a
symmetric circulant matrix and factorized via the FFT. We are interested in realizations at equidistant times 6k, k = 0, . . . , n, for some 6 > 0, and let Xk= X(6k), for a
process with mean ft. Suppose that the first column of the covariance matrix E is given
by s = (Uo,..
.,)T. This
(n+1) x (n+1) matrix can be embedded in a 2n x 2m symmet-
ric circulant matrix C, where the first column is given by c= (O-O,
... , O-,
on-,
...
,1)T
(remaining columns are obtained by circularly permuting the indices). The fundamental fact is that the discrete Fourier transform diagonalizes the symmetric circulant
matrix C. If we let F be the 2n x 2n discrete Fourier matrix (with inverse F
= F/2n,
where the overline is the complex conjugate), then the vector of eigenvalues of C are
given by A = Fc, or equivalently A = Fc since C is symmetric, and we have the
fact that C = F- 1 AF, where A = diag(A). Therefore, if we let D =F diag(A/2'n),
then DD
= F diag(A)F/2n = FAF-
1
= C, and so D is a (complex) square root
factorization of C. Using this factorization the adapted version of Algorithm A.1 is
given in the following algorithm, which generates two realization X 1 , X 2 in O(n log n)
floating point operations.
Algorithm A.2 (Fast Stationary Gaussian Process Simulation)
1. Compute the eigenvalues A = Fc via the FFT.
2. Compute the vector Yik =
Ak/2n.
3. Generate Y1,1..., Y, Y1 2 ,y'
(Y..
yi)Ti
~N(0, 1), and let Z =Y+iY 2 , where Y=
1 2.
4. Compute the vector C with entries (k
FFT.
62
=
Z%l, and compute V = FC, via the
5. Let A be the first n + 1 components of V.
6. Output X 1
=
p + Re(A) and X 2 =p + Im(A).
63
64
Bibliography
[1] A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations. New York: WileyInterscience, 1984.
[2] J. Pedlosky, Ocean Circulation Theory. Springer-Verlag, 1998.
[3] R. Salmon, Lectures on Geophysical Fluid Dynamics. Oxford University Press,
1998.
[4] T. DelSole, "Stochastic models of quasigeostrophic turbulence," Surv. Geophys.,
vol. 25, pp. 107-149, 2004.
[5] A. J. Majda, R. V. Abramov, and M. J. Grote, Information Theory and Stochastics
for Multiscale Nonlinear Systems, vol. 25 of CRM Monograph Series. American
Mathematical Society, 2005.
[6] A. J. Majda, D. W. McLaughlin, and E. G. Tabak, "A one-dimensional model
for dispersive wave turbulence," J. Nonlinear Sci., vol. 6, pp. 9-44, 1997.
[7] K. Dysthe, H. Krogstad, and P. Muller, "Oceanic rogue waves," Annu. Rev. Fluid
Mech., vol. 40, p. 287, 2008.
[8] W. Xiao, Y. Liu, G. Wu, and D. K. P. Yue, "Rogue wave occurrence and dynamics by direct simulations of nonlinear wave-field evolution," Journal of Fluid
Mechanics, vol. 720, pp. 357-392, 2013.
[9] A. J. Majda and M. Branicki, "Lessons in uncertainty quantification for turbulent dynamical systems," Discrete and Continuous Dynamical Systems, vol. 32,
pp. 3133-3221, 2012.
[10] M. Branicki and A. J. Majda, "Quantifying uncertainty for predictions with model
error in non-gaussian systems with intermittency," Nonlinearity, vol. 25, p. 2543,
2012.
[11] A. J. Majda and J. Harlim, Filtering Complex Turbulent Systems. Cambridge
University Press, 2012.
[12] W. Cousins and T. P. Sapsis, "Quantification and prediction of extreme events in a
one-dimensional nonlinear dispersive wave model," Physica D, vol. 280, pp. 48-58,
2014.
65
[13] N. Chen, A. J. Majda, and D. Giannakis, "Predicting the cloud patterns of
the madden-julian oscillation through a low-order nonlinear stochastic model,"
Geophysical Research Letters, vol. In Press, 2014.
[14] H. W. Broer, I. Hoveijn, and M. v. Noort, "A revertible bifurcation analysis of
the inverted pendulum," Physica D, vol. 112, pp. 50-63, 1997.
[15] D. Shadman and B. Mehri, "A non-homogeneous hill's equation," Applied Mathematics and Computation, vol. 167, pp. 68-75, 2005.
[16] P. D. Kourdis and A. F. Vakakis, "Some results on the dynamics of the linear
parametric oscillator with general time-varying frequency," Applied Mathematics
and Computation, vol. 183, pp. 1235-1248, 2006.
[17] L. Arnold, I. Chueshov, and G. Ochs, "Stability and capsizing of ships in random
sea - a survey," Nonlinear Dynamics, vol. 36, pp. 135-179, 2004.
[18] M. Gitterman, The Noisy Oscillator: The first hundred years, from Einstein until
now. New Jersey, USA: World Scientific, 2005.
[19] A. Babiano, J. H. Cartwright, 0. Piro, and A. Provenzale, "Dynamics of a small
neutrally buoyant sphere in a fluid and targeting in hamiltonian systems," Physical Review Letters, vol. 84, no. 25, p. 5764, 2000.
[20] T. P. Sapsis and G. Haller, "Instabilities in the dynamics of neutrally buoyant
particles," Physics of Fluids, vol. 20, p. 017102, 2008.
[21] G. Haller and T. P. Sapsis, "Localized instability and attraction along invariant
manifolds," SIAM J. of Appl. Dynamical Systems, vol. 9, pp. 611-633, 2010.
[22] J. C. Poggiale and P. Auger, "Impact of spatial heterogeneity on a predator-pray
system dynamics," C.R. Biol., vol. 327, pp. 1058-1063, 2004.
[23] A. Naess and T. Moan, Stochastic Dynamics of Marine Structures. Cambridge
University Press, 2012.
[24] T. Soong and M. Grigoriu, Random Vibration of Mechanical and Structural Systems. PTR Prentice Hall, 1993.
[25] A. Naess, "Extreme value estimates based on the envelope concept," Applied
Ocean Research, vol. 4, no. 3, p. 181, 1982.
[26] M. Nicodemi, "Extreme value statistics," in Encyclopedia of complexity and systems science, vol. E, p. 3317, 2009.
[27] M. R. Leadbetter, G. Lindgren, and H. Rootzen, Extremes and related properties
of random sequences and processes. Springer, New York, 1983.
[28] R.-D. Reiss and M. Thomas, Statistical analysis of extreme values. Birkhduser;
3rd edition, 2007.
66
-1
11
11.1
-11-1
_--
1-1-. l---
,44O.Wu&u.,AA -,.-
.-. ,-.
-.
ta,
--
_-'__-_
"
. --
,
-
--
,
1..
[29] P. Embrechts, C. Kluppelberg, and T. Mikosch, Modeling Extremal Events.
Springer, 2012.
[30] J. Galambos, The asymptotic theory of extreme order statistics. Wiley Series in
Probability and Statistics, 1978.
[31] E. Kreuzer and W. Sichermann, "The effect of sea irregularities on ship rolling,"
Computing in Science and Engineering, vol. May/June, pp. 26-34, 2006.
[32] K. Sobczyk, Stochastic Differential Equations. Dordrecht, The Netherlands:
Kluwer Academic Publishers, 1991.
[33] T. P. Sapsis and G. A. Athanassoulis, "New partial differential equations governing the joint, response-excitation, probability distributions of nonlinear systems,
under general stochastic excitation," Probab. Eng. Mech., vol. 23, no. 2-3, pp. 289
- 306, 2008.
[34] D. Venturi, T. P. Sapsis, H. Cho, and G. E. Karniadakis, "A computable evolution
equation for the joint response-excitation probability density function of stochastic
dynamical systems," Proceedings of the Royal Society A, vol. 468, p. 759, 2012.
[35] I. F. Blake and W. C. Lindsey, "Level-crossing problems for random processes,"
IEEE Trans. Information Theory, vol. IT-19, pp. 295-315, 1973.
[36] M. F. Kratz, "Level crossings and other level functionals of stationary Gaussian
processes," Probab. Surv., vol. 3, pp. 230-288, 2006.
[37] S. 0. Rice, "Distribution of the duration of fades in radio transmission: Gaussian
noise model," Bell System Tech. J., vol. 37, pp. 581-635, 1958.
[38] A. Champneys, "Dynamics of parametric excitation," in Encyclopedia of Complexity and Systems Science (R. A. Meyers, ed.), pp. 2323-2345, Springer New
York, 2009.
[39] Y. K. Lin and C.
1995.
Q.
Cai, ProbabilisticStructural Dynamics. McGraw-Hill, Inc.,
[40] R. Berthet, A. Petrosyan, and B. Roman, "An analog experiment of the parametric
instability," American Journal of Physics, vol. 70, no. 7, 2002.
[41] D. R. Rowland, "Parametric resonance and nonlinear string vibrations," American
Journal of Physics, vol. 72, no. 6, 2004.
[42] L. Arnold, I. Chueshov, and G. Ochs, "Random dynamical systems methods in
ship stability: A case study," in Interacting Stochastic Systems (J.-D. Deuschel
and A. Greven, eds.), pp. 409-433, Springer Berlin Heidelberg, 2005.
67
[43] V. Belenky, A. Reed, and K. Weems, "Probability of capsizing in beam seas with
piecewise linear stochastic gz curve," in Contemporary Ideas on Ship Stability
and Capsizing in Waves (M. Almeida Santos Neves, V. L. Belenky, J. 0. de Kat,
K. Spyrou, and N. Umeda, eds.), vol. 97 of Fluid Mechanics and Its Applications,
pp. 531-554, Springer Netherlands, 2011.
[44] V. Belenky, K. Weems, W.-M. Lin, and J. Paulling, "Probabilistic analysis of
roll parametric resonance in head seas," in Contemporary Ideas on Ship Stability
and Capsizing in Waves (M. Almeida Santos Neves, V. L. Belenky, J. 0. de Kat,
K. Spyrou, and N. Umeda, eds.), vol. 97 of Fluid Mechanics and Its Applications,
pp. 555-569, Springer Netherlands, 2011.
[45] K. Spyrou, "Simple analytical criteria for parametric rolling," in Contemporary
Ideas on Ship Stability and Capsizing in Waves (M. Almeida Santos Neves, V. L.
Belenky, J. 0. de Kat, K. Spyrou, and N. Umeda, eds.), vol. 97 of Fluid Mechanics
and Its Applications, pp. 253-266, Springer Netherlands, 2011.
[46] J. B. Roberts and M. Vasta, "Markov modelling and stochastic identification for
nonlinear ship rolling in random waves," PhilosophicalTransactions of the Royal
Society of London A: Mathematical, Physical and Engineering Sciences, vol. 358,
no. 1771, pp. 1917-1941, 2000.
[47] L. Dostal, E. Kreuzer, and N. Sri Namachchivaya, "Non-standard stochastic averaging of large-amplitude ship rolling in random seas," Proceedings of the Royal
Society of London A: Mathematical, Physical and Engineering Sciences, vol. 468,
no. 2148, pp. 4146-4173, 2012.
[48] M. S. Longuet-Higgins, "On the joint distribution of the periods and amplitudes
of sea waves," Journal of Geophysical Research, vol. 80, no. 18, pp. 2688-2694,
1975.
[49] C. M. Bender and S. A. Orszag, Advanced mathematical methods for scientists
and engineers. I Springer-Verlag, New York, 1999. Asymptotic methods and
perturbation theory, Reprint of the 1978 original.
[50] R. Langley, "On various definitions of the envelope of a random process.," Journal
of Sound and Vibration, vol. 105, no. 3, pp. 503 - 512, 1986.
[51] D. J. Higham, "An algorithmic introduction to numerical simulation of stochastic
differential equations," SIAM Review, vol. 43, no. 3, pp. 525-546, 2001.
[52] D. B. Percival, "Simulating gaussian random processes with specified spectra,"
Computing Science and Statistics, vol. 24, pp. 534-538, 1992.
[53] B. Gershgorin, J. Harlim, and A. J. Majda, "Test models for improving filtering
with model errors through stochastic parameter estimation," Journal of Computational Physics, vol. 229, no. 1, pp. 1 - 31, 2010.
68
[54] B. Gershgorin, J. Harlim, and A. J. Majda, "Improving filtering and prediction
of spatially extended turbulent systems with model errors through stochastic
parameter estimation," Journal of Computational Physics, vol. 229, no. 1, pp. 32
-
57, 2010.
[55] M. Branicki, B. Gershgorin, and A. J. Majda, "Filtering skill for turbulent signals
for a suite of nonlinear and linear extended kalman filters," J. Comput. Phys.,
vol. 231, pp. 1462-1498, 2012.
[56] N. Chen, D. Giannakis, R. Herbei, and A. J. Majda, "Au mcmc algorithm for
parameter estimation in signals with hidden intermittent instability," SIA M/ASA
Journal on Uncertainty Quantification, vol. 2, no. 1, pp. 647-669, 2014.
[57] F. C. Adams and A. M. Bloch, "Hill's equation with random forcing terms,"
SIAM Journal on Applied Mathematics, vol. 68, no. 4, pp. 947-980, 2008.
[58] C. Elachi, "Waves in active and passive periodic structures: A review," Proceedings
of the IEEE, vol. 64, pp. 1666-1698, Dec 1976.
[59] K. Kumar and L. S. Tuckerman, "Parametric instability of the interface between
two fluids," Journal of Fluid Mechanics, vol. 279, pp. 49-68, 11 1994.
[60] T. B. Benjamin and F. Ursell, "The stability of the plane free surface of a
liquid in vertical periodic motion," Proceedings of the Royal Society of London A:
Mathematical, Physical and EngineeringSciences, vol. 225, no. 1163, pp. 505-515,
1954.
[61] F. J. Poulin and G. R. Flierl, "The stochastic mathieu's equation," Proceedings of
the Royal Society of London A: Mathematical, Physical and Engineering Sciences,
vol. 464, no. 2095, pp. 1885-1904, 2008.
[62] K. Lindenberg and B. J. West, The nonequilibrium statisticalmechanics of open
and closed systems. New York: VCH Publishers, c1990., 1990.
[63] C. Floris, "Stochastic stability of damped mathieu oscillator parametrically excited by a gaussian noise.," Mathematical Problems in Engineering, vol. 2012,
pp. 1 - 18, 2012.
[64] D. P. Kroese and Z. I. Botev, "Spatial Process Generation," ArXiv e-prints, Aug.
2013.
[65] C. R. Dietrich and G. N. Newsam, "A fast and exact method for multidimensional gaussian stochastic simulations," Water Resources Research, vol. 29, no. 8,
pp. 2861-2869, 1993.
69