Marginal Instability and Intermittency
in Stochastic Systems
M.F. Dimentberg1, A. Hera2 and A. Naess3
1
Mechanical Engineering Department, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, USA
2
Information Technology Division, Worcester Polytechnic Institute, 100 Institute
Road, Worcester, MA 01609, USA
3
Centre for Ships and Ocean Structures and Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
Abstract. Dynamic systems with lumped parameters which experience random temporal variations are considered. The variations may “smear” boundary between the system’s states which are dynamically stable and unstable in the classical sense. The system’s response within such a “twilight
zone” of marginal instability is found to be of an intermittent nature, with alternating periods of zero or
almost zero response and rare short outbreaks. As long as it may be impractical to preclude completely
such outbreaks for a designed system, the corresponding response should be analyzed to evaluate the
system’s reliability.
Results of such analyses are presented separately for cases of slow and rapid parameter variations.
Linear models of the systems are studied in the former case using parabolic approximation for the variations in the vicinity of their peaks together with Krylov-Bogoliubov averaging for the transient response. This results in a solution for the response probability density function (PDF). The analysis is
also used to derive on-line identification procedure for the system from its observed response with set
of rare outbreaks. Potential examples of applications include 1D and 2D short-term galloping of elastically suspended bodies in cross-flow of fluid with random temporal variations of flow speed; bundles
of heat exchanger tubes in cross-flow with potential for flutter-type instability; and rotating shafts.
The case of rapid broadband parameter variations is studied using theory of Markov processes. The
system is assumed to operate beyond its stochastic instability threshold – although only slightly – and
its nonlinear model is used accordingly. The analysis is based on solution of the Fokker-PlanckKolmogorov (FPK) partial differential equation for stationary PDF of the response. Several such PDFs
are analyzed; they are found to have integrable singularities at the origin indicating an intermittent nature of the response. One of potential applications is population dynamics where behaviour of predatorprey (or parasite-host) pair in random environment is studied using extended stochastic Lotka-Volterra
model. The analysis provides potential for probabilistic predictions of response outbreaks, in particular
for the cases of intermittency (like the notorious case of seven outbreaks in budworms (forest parasites)
in eastern Canada since 1710).
Key words: marginal instability, excursions, probability density function, intermittency
1
2
M.F. Dimentberg, A.Hera and A.Naess
1 Introduction
Classical definitions of stability and instability deal with long-term behavior of
dynamic systems, that is, behavior as time t   . These definitions are known to
be not perfectly appropriate for applications with limited service life (such as missiles, projectiles, etc.) which may sometimes be qualified as acceptable in spite of
being unstable in the classical sense. Design of such marginally unstable systems
may be based on analysis of their transient response within limited service life.
These classical definitions may also prove to be not perfectly adequate for
some dynamic systems that may be intended for long-term operation. Such systems are designed, as a rule, to operate within their stability domain in the classical sense as long as their “nominal” design parameters are considered. However, if
the parameters may experience random temporal variations around their “nominal” or expected values, the system may become “marginally unstable” within the
“smeared” stability boundary. Whenever complete elimination of this kind of response may lead to impossible or impractical design the corresponding short-time
outbreaks in response should be analyzed to evaluate the system’s reliability with
respect to, say, first-passage failure and/or of low-cycle fatigue. Relevant dynamic
studies may also be of importance for interpretation of measured response signals.
Results of such analyses are presented separately for cases of slow and rapid
parameter variations. Linear models of the systems are studied in the former case
using parabolic approximation (PA) for the variations in the vicinity of their peaks
together with Krylov-Bogoliubov (KB) averaging for the transient response. This
results in a solution for the probability density function (PDF) of the response in
terms of that of the bifurcation parameter. The analysis is also used to derive procedure for on-line system identification from its observed intermittent response.
The case of rapid broadband parameter variations is studied using theory of
Markov processes. The system is assumed to operate beyond its stochastic instability threshold but only slightly and its nonlinear model is used accordingly. The
analysis is based on solution of the FPK equation for stationary response PDF.
Several such PDFs are analyzed with integrable singularities at their origins indicating an intermittent nature of the response. One of the potential applications is
population dynamics where behaviour of predator-prey pair in random environment is studied using extended stochastic Lotka-Volterra model.
2 Systems with slow variations of parameters
Linear systems with slow stationary random temporal variations of parameter(s)
are considered which operate within classical stability domain for the mean or
“nominal” system. However, any brief excursion beyond the instability threshold
may lead to growth of the system’s response. The growth is assumed to be limited
as long as the system quickly returns back into the stability domain. The response
Marginal Instability and Intermittency in Stochastic Systems
3
would be seen then as a set of spontaneous brief outbreaks alternating with intervals of zero response.
The Slepian model [12] of a stationary zero-mean random process g(t) with
unit standard deviation implies parabolic approximation (PA) [14] in the vicinity
of its peak –during upcrossing given level u that starts at time instant t = 0 namely
g  t u   u  1 u   t   2t 2 2  so that g  t   u   t   u 2  t 
2
for t  0, 2  2 u  and max t g  t   g   2u   g p  u   2 2 2u .
(1)
Subscript “p” is used for peak values of random processes,  is random slope
of g(t) at the instant of upcrossing and  2   g2 

     d 
2
gg
where  gg  

is power spectral density (PSD) of g(t) so that  is a mean frequency of g(t).
Thus, according to the equation (1) random process g(t) is regarded as deterministic within the high-level excursion of duration  f   t f  2   u above level u,
depending just on its initial slope  at the instant of upcrossing. This slope is regarded as a random variable for the excursion; in particular, it has the Rayleigh
PDF in case of a Gaussian g(t)[14] . This probabilistic description is used together
with the solution for the transient response within the instability domain.
The first example is a SDOF system with randomly varying damping – say,
Den-Hartog model of 1D galloping [2] under variable windspeed
X  2   q  t   X   2 X  0 so that q  t    q  g t  and u    q .
(2)
Substituting the PA (1) into equation (2) reduces the latter to an ordinary differential equation (ODE) with a single random parameter  . This ODE for a certain representative crossing should be integrated starting from the instant of upcrossing tu until instant of peak of X(t) for a given outbreak. Here it can be done
analytically using the KB-averaging over the response period for a quite common
case of a lightly damped system (2) with slow temporal variations of the damping
coefficient:   q  t    ,    [1]. The method leads to first-order ODE for
slowly varying amplitude A  t   X 2  X 2 2 which has the solution
A    A0 exp f   ; f     q       2 2   u 3 6 ,    t  tu  .
The peak amplitude of the response as attained at  f  2   u is
(3)
4
M.F. Dimentberg, A.Hera and A.Naess
Ap  A  f   A0 exp  2  ,    q 3u 2     .
3
(4)
This solution together with equation (1) define, in parametric form, relation between Ap  Ap A0 and g p . Let Ap  h( g p ) for g p  u . Then the function inverse
to h (denoted by superscript “-1”) can be obtained as
2
g p  u     1 2u   h1  Ap   u  1 2u   3u 2 2 q  ln Ap  .
23
(5)
These relations open way to predicting reliability for the system (2) based on
relevant statistics of g(t). Thus, the first-passage problem for A(t) with barrier A* is
reduced to that for g(t) with barrier g*  h 1  A*  as evaluated by equation (5).
Furthermore, the PDF of g(t) can be used to obtain the PDF of Ap as


p  Ap   pg h1  Ap   dh1 dAp .
(6)
This PDF is non-zero for Ap  1 rather than for Ap  0 and is normalized not to
unity but to Pr ob  g p  u ; according to the equations (5) and (6) it has a singularity at Ap  1 .
Figure 1 illustrates response sample of the system (2) with   0.16,   2
which contains one excursion of the apparent damping into negative domain (see
dash-dot curve of q(t)) with the corresponding response outbreak. To guarantee
nonzero response during the short-term instability a small zero-mean stationary
broadband random process had been added to the RHS of the equation (2); one
can see that the corresponding subcritical response is really very small.
Fig. 1 Response sample with “outbreak” (solid line) of a SDOF system with apparent linear viscous damping 0.16 – q(t); sample of random process q(t) is shown by dotted line.
Marginal Instability and Intermittency in Stochastic Systems
5
In Fig. 2 the PDF p  Ap  for the case of Gaussian g(t) is compared with Monte-Carlo simulation (where values of A0 were measured for each upcrossing). The
corresponding prediction of the PDF of the actual (nonscaled) response amplitude
and/or its peaks may be improved if PDF p  A0  of the random variable A0 is
known [8].
Amplitude pdf
0.25
0.2
0.15
p( Ap )
0.1
0.05
0
1
2
3
4
5
6
Ap
Ap  Ap / A0 and corresponding histogram as ob-
A0 pdf
Fig. 2 Theoretical
0.5PDF of scaled amplitude
tained from sample of X(t).
0.4
The simplest approach for getting rough estimates for A0 is just to ignore the
0.3
parameter(s) variations in prediction of small-level steady-state response [7, 8].
However, such0.2a crude approximation may be unconservative. To improve its accuracy a model with deterministic parameter variations and external random exci0.1
tation is considered.
0
4
X  2   q(t )  0X  2 X2  t  with
q t  6  sin t8 and  , 10
 , q,   .
(7)
To this model the method of moments
can
aPeak
pdfbe applied together with stochastic av0.1
eraging [4, 13] resulting in three ODEs for second-order moments Dcc , ss  X c2, s ,
0.08
Dcs  X cs where
X c  t 0.06
 X cos t  ( X )sin t , X s t   X sin t  ( X )cos t.
Numerical solutions
to this ODE set have been obtained within  0, tu  with initial
0.04
conditions (ICs) corresponding to steady-state response of a system with constant
2
parameters: D0.02
. The results provide ICs for the
cc  0  Dcc  0  D 4 , Dcs  0   0
0
following analytical
solution
as 4based on
the PA
for q10
t    sin t within
0
2
6
8
tu , t f  ,  tu  sin 1    ,  t f    sin 1   
6
M.F. Dimentberg, A.Hera and A.Naess

2
Dcc , ss  t f   Dcc , ss  tu   exp  4 3       1


32

.
(8)
The results may be compared with “stationary approximation” whereby the simple
2
ICs are enforced as Dcc, ss tu   D 4 . Figure 3 illustrates the corresponding
correction factor for peak values of D  t   Dcc t   Dss t  as established upon
identifying  as peak value of g(t):

K  1/ 
1
D ( z )numerical
D ( z )stationary
p( z *)dz, p  z *  u 2 z  exp   u 2 z 2 2  / P* for z  1
where z     g p u , g p  q p  q , u    q .
1.35
u= 1
u=1.5
u= 2
u= 3
u= 4
Correction factor
1.3
1.25
1.2
1.15
1.1
1.05
1
2
3
4
5
6
7
8
9
10
/
Fig. 3 Correction factor K as function of / and u.
These correction factors may be used to obtain improved estimates for initial
values of the response amplitude at each starting instant of the response outbreak
due to short-term instability. Namely, the estimate of a steady-state rms value for
the system without parameter variations may be directly multiplied by K . As
could be expected, the correction is seen to be reduced with increasing  .
The equation (4) is convenient for evaluating the system’s properties from its
measured (on-line!) response with outbreaks as one shown in Fig. 1. To this end
one can use peak amplitudes Ap , as attained at instants  f  2   u in the local
time frames and corresponding amplitudes Ai at inflexion points of the curve
ln A   . From the equation (4)
Ai  A  i   A0 exp  , so that Ap Ai  exp 
and A0  A Ap Thus, for each one of the observed response outbreaks one can
2
i
identify in a global time frame the instants t f  tu   f  and ti  tu   i  which
Marginal Instability and Intermittency in Stochastic Systems
7
correspond to peak and inflexion-point amplitudes Ap and Ai respectively; the in-
stants of upcrossings can also be identified as tu  t f  2  t f  ti   2ti  t f . The
frequency  may now be obtained by averaging time difference t f  ti over all observed outbreaks of response. The identification procedure as described in details
in [7] relies upon averaging ln( Ap Ai )  exp  over all observed outbreaks. It provides on-line estimates both for the mean apparent damping coefficient – which
may be regarded as a nominal stability margin – and for standard deviation and
mean frequency of its random temporal variations.
The described analytical solutions can be extended to certain TDOF systems
with certain “symmetry” which permit to “wrap up” two equations of motion into
a single complex equation [6, 8]. Thus, translational and tilting oscillations of the
Jeffcott rotor had been considered in [6] and [8] respectively.
In the general case of TDOF system two coupled response amplitudes remain
after KB-averaging for the case of lightly damped system thereby requiring numerical integration for the two ODEs of slow motion from starting point of the response outbreak till the instant when both response variables pass their peaks.
Then, as long as relation is established (numerically) between peak value(s) of response(s) and that of scaled zero-mean part g(t) of the bifurcation parameter the
basic procedure can be applied for predicting response PDF. Several examples of
such analysis as presented in [8] are:
 Rotating shaft with anisotropic stiffness in translational vibrations;
 TDOF flutter of a tube row in heat exchanger in cross-flow of fluid;
 Two-dimensional galloping of a rigid body in a fluid flow (case of full
2  2 damping matrix).
3 Systems with rapid variations of parameters
Systems with broadband stationary random temporal variations of parameters are
considered here which may be described by the theory of Markov processes. They
have clearly defined boundaries corresponding to various definitions of stochastic
stability for the system’s linear part [13]. They operate within the domain of stochastic instability - although close to the corresponding instability threshold where the instability may be called marginal indeed. Therefore, adequate modelling requires the system’s nonlinearity to be accounted for. The response is found
to be of the intermittent nature indeed in such cases. The analysis provides the potential for predicting the response PDF’s through solution of the stationary Fokker-Planck-Kolmogorov (FPK) partial differential equation. Several such stationary PDF’s are analyzed; all of them are found to possess an integrable singularity
at the origin, whereas the response itself does exhibit the intermittency indeed.
Common characteristic features of these solutions are also certain other typical
patterns of a stationary intermittent response; for example, if amplitude A(t) has
8
M.F. Dimentberg, A.Hera and A.Naess
mean <A> and standard deviation  A then typically a small relative stay time of
A(t) above <A> is observed, and also  A >> <A>.
Stationary PDF of response amplitude with singularity at zero has been
known for a long time for a SDOF system with small nonlinear damping [14] but
correlation between the singularity and intermittency has been established by numerical simulation only recently [11]. Extensive study of intermittency in distributed-parameter system through their response moments see in [10].
Solutions to the FPK equations for response PDFs are presented here for two
cases of potential intermittency in systems with impacts. They have different types
of damping than in [11, 14] which may effectively restrict response growth in case
of stochastic instability. The first is that of a single-barrier SDOF system with inelastic impacts as governed by the equation of motion between impacts for its displacement Y(t)
Y  2 Y   2Y 1    t    0 for Y  h ,
(9)
and impact condition at the barrier at Y = -h (with subscripts “plus” and “minus”
corresponding to rebound and impact velocities respectively)
Y  rY where Y  Y t*  0 , Y t*   h and 0  r  1 .
(10)
Here r is a restitution factor,   t  is a stationary zero-mean Gaussian white noise
with intensity D . Stationary PDF of response energy H  t   1 2   X 2   2 X 2 
had been obtained in [3] for the system (9), (10) by a quasiconservative stochastic
averaging [4, 12]. It does exist (is normable) if and only if
  1 and  1  r    1     where    2B and is described by different
analytical expressions for H  H h and for H  H h where H h   2 h 2 2 is the
system’s potential energy at the barrier. The first of these expressions which may
be regarded as the conditional PDF pc  H   p( H H  H h ) normalized within
[0, H h ] is
pc  H   1    H h  ( H H h )  .
(11)
The (conditional) mean and relative stay time above this mean of the response
energy may be found from (11) as
Marginal Instability and Intermittency in Stochastic Systems
H  H h 1     2    , H 
Hh

pc  H  dH
H
H

0
9
1
 2  
pc  H  dH  

 1  
 1.
Thus H << 1 for small 1   clearly indicating that the response should be intermittent indeed in this case. The intermittency may also be seen from expected time
<T> for reaching mean energy from initial state with H  H . The latter may be
adopted as an index of expected period between outbreaks and can be obtained
from solution to the corresponding Pontryagin equation for T(H) - one with given
initial state H [13]. Averaging this T(H) over H using the conditional PDF (11)
yields
H
 T 
 

 T  H  p  H  dH   21     2   
 1
 1 
c
 1  2 1     . (12)
0
This expected period between outbreaks is clearly increasing with reducing 1  
The other case of intermittency is considered for vibroimpact system with doublesided barrier and linear damping. The equation of motion between impacts is
Y  2 Y 1    t     2Y 1    t    0 for  h  Y  h ,
(13)
where   t  ,  t  are stationary zero-mean independent Gaussian white noises with
intensities D , D   2   D respectively. Impact condition (10) with r = 1 is
2
now imposed both at Y = - h and Y = + h. Under the above relation between intensities of excitations the FPK equation for the joint PDF of response displacement
and velocity has an exact stationary solution [4]
w  y, y   C
y
2
 y 2 2  where   2 D 2  1 2 .

(15)
Without barriers this solution is not normable. However, nonlinearity due to the
double-sided barrier may restrict response growth even in the absence of nonlinear
damping if 1 2    1 . Specifically, integrating PDF (15) over y provides PDF
p(y) of the displacement
p  y 
1 
h  y h
2 1
for  h  y  h
(16)
with integrable singularity at the origin if   1 2 . Relative stay time  Y of the
magnitude of Y(t) above its mean may be considered as an index of intermittency
and it is seen to approach zero with 1    0 thereby indicating intermittency:
10
M.F. Dimentberg, A.Hera and A.Naess
h
Y  2  yp  y  dy 
2 1   
0
 2 1     3  2  
3  2
2 1 
 h, P  Pr ob Y  Y   2
Y
 p  y  dy 
0
(17)
and  Y  1  P  P .
Finally, consider extended stochastic Lotka-Volterra (L-V) model [4, 5] of interacting populations of the predator-prey or parasite-host type
u  mu  k  uv, v   v 1    t     uv   v 2 .
(18)
Here u(t) and v(t)are population sizes of predators (or parasites) and preys (or
hosts) respectively, whereas   t  is a zero-mean Gaussian random white noise
with intensity D. The system (18) has asymptotically stable equilibrium point
u0     m k   /  ; v0  m k  and an additional equilibrium state u*  0,
v*    which is unstable if vo  v* or    *   k  m . At the bifurcation point
   * these two equilibrium states merge, with the state u*  0, v*    becoming stable for    * . Physical meaning of these transformations: beyond this transcritical bifurcation point growth of the preys’ population is bounded by its interspecies competition rather than by predators’ activity, whereas the latter become
extinct because of food shortage. Joint stationary PDF w(u, v) of population sizes
as derived in [4, 5] from exact solution to the corresponding FPK equation is
w  u, v   p  u k   p  v  , where p  z   z zo 1e z   zo  and   2 D 2
(19)
where Г is gamma-function and z  u k ,z0  u0 k and z  v,z0  v0 for stationary PDFs of u(t) and v(t), respectively. Thus, both steady-state population
sizes u(t) and v(t) are independent gamma-distributed random variables with the
following mean values and standard deviations
u  u0     m k       1  vo v*  , v  v0  m k 
and  u 
u  u 
2
 uo k  ,  v 
v  v 
2
 vo  .
(20)
Both these PDFs do exist provided that    * . It can be seen from the equations (19) that intermittent behavior should be expected for v(t) (u(t)) if
v0  1 ( u0 k  1), so that  v  vo ( u  uo ) , with rare and short pulse-like
intensive outbreaks in v(t) (u(t)) and low-level oscillations between the pulses.
Relevant indices of intermittency such as small relative stay time above the mean
and large values of the ratio standard deviation/mean may also be used. Figure 4
Marginal Instability and Intermittency in Stochastic Systems
11
illustrates such a behavior of predators for small 1  vo v*  1    * . Cases with
intermittent behavior of both predators and preys are illustrated in [5, 9].
1.5
u
1
0.5
50
100
150
200
time
250
300
350
400
50
100
150
200
time
250
300
350
400
4
3
v
2
1
0
Fig. 4 Samples of u(t) and v(t) as obtained from numerical simulation for the system (18) with
m=1, =1, =1, k=1, =0.98, D=1. It illustrates an intermittency in predators only (u0=.01 and
=1.98). All quantities are non-dimensional.
4 Conclusions
Temporal random variations of parameters in dynamic systems may “smear” classical neutral stability boundaries. The system’s response within such a “twilight
zone” of marginal instability is found to be of an intermittent nature, with alternating periods of zero (or almost zero) response and rare short outbreaks. As long
as it may be impractical to preclude completely such outbreaks for a designed system its response should be analyzed to evaluate reliability, in particular, to predict
response PDF and/or to solve first-passage problem.
Procedures and results of such analyses are presented for cases of slow and rapid parameter variations. Linear models may be adequate in the former case where
using parabolic approximation for the variations together with transient analysis
yields response PDF. The analysis is also used to derive on-line identification procedure for the system from its observed response with set of rare outbreaks. Potential applications include 1D and 2D short-term galloping of suspended bodies in
fluid cross-flow; heat exchanger tubes in cross-flow with potential flutter-type instability; rotating shafts.
The case of broadband parameter variations is studied using theory of Markov
processes with the system operating beyond its stochastic instability threshold –
although only slightly – and its nonlinear model is used accordingly. The analysis
is based on solution of the FPK equation for stationary PDF of the response. Several such PDFs are analyzed; they are found to have integrable singularities at the
origin indicating an intermittent nature of the response. One of the potential appli-
12
M.F. Dimentberg, A.Hera and A.Naess
cations is population dynamics where behaviour of predator-prey pair in random
environment is studied using extended stochastic L-V model. The analysis provides potential for probabilistic predictions of response outbreaks, in particular for
the cases of intermittency (like the notorious case of seven outbreaks in budworms
(forest parasites) in eastern Canada since 1710).
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