Normal Modes of Coupled Oscillators
The Lorenz Attractor
Extra Special Bonus Material
Nick Fisher
Department of Mathematics and Statistics
Colorado School of Mines
Tuesday 2nd December, 2014
Normal Modes of Coupled Oscillators
Table of contents
1
Normal Modes of Coupled Oscillators
Linear Spring-Mass System
Dark Soliton Interactions
2
The Lorenz Attractor
The Lorenz Equations
Separation of Initial Conditions
Volume Contraction in Phase Space
Bifurcations
Transient Chaos
The Lorenz Attractor
Normal Modes of Coupled Oscillators
The Lorenz Attractor
Coupled Spring-Mass System: Schematic
Figure: The schematic of a coupled spring-mass system.
Normal Modes of Coupled Oscillators
The Lorenz Attractor
Coupled Spring-Mass System: Equations of Motion
In what follows, we ignore any effects due to damping. Then, the
equations of motion for the coupled spring-mass system can be
derived by applying Hooke’s law, and Newton’s Second Law.
m1 ẍ1 = −k1 x1 + k2 (x2 − x1 )
(1)
m2 ẍ2 = −k2 (x2 − x1 ) + k3 (x3 − x2 )
(2)
m3 ẍ3 = −k4 x3 + k3 (x2 − x3 )
(3)
Normal Modes of Coupled Oscillators
The Lorenz Attractor
Coupled Spring-Mass System: Normal Modes
In matrix form, the equations of motion can be written as:
Ẍ = AX
(4)
where,
 −(k2 +k1 )

A=
m1
k2
m2
0
k2
m1
−(k3 +k2 )
m2
k3
m3
0


k3

m2
−(k4 +k3 )
m3
(5)
In order to find the “normal modes” of the equations of motion,
we seek solutions of the form Ẍ = ω 2 X. That is, by solving the
eigenvalue problem (A − ω 2 I )X = 0, the eigenvectors will yield the
modes, while the associated eigenvalues will describe their
frequencies of oscillation.
Normal Modes of Coupled Oscillators
The Lorenz Attractor
Coupled Spring-Mass System: Example
Suppose m1 = m2 = m3 = 1 , k1 = k4 = k2 = k3 = 1. By
plugging these values in to equation (5), we find the following:


−2 1
0
A =  1 −2 1 
(6)
0
1 −2
√
√
Thus, the eigenvalues are ω 2 = {−2 − 2, −2 + 2, −2} with the
corresponding eigenvectors (modes)

    
1
1 

√
√1
− 2 ,  2 ,  0 
(7)


−1
1
1
Normal Modes of Coupled Oscillators
The Lorenz Attractor
Dark Soliton Interactions
Figure: Dark solitons on a homogeneous background.
Dark solitons are a type of non-linear wave. They are of special
interest to the study of Bose-Einstein condensates.
Normal Modes of Coupled Oscillators
The Lorenz Attractor
Dark Soliton Interactions: Equations of Motion
The distribution of of a Bose-Einstein condensate in a magnetic
trap is described by the non-linear Schrödinger equation:
1
i ut + uxx − |u|2 u = V (x)u
2
(8)
where V (x) = Ω2 x 2 describes the strength of the trap.
Fortunately, the motion of three interacting dark solitons can be
described by a system of ODE’s:
3/2
√
2
ẍ1 = −ωosc
x1 − 8n0 e −2
3/2
2
ẍ2 = −ωosc
x2 + 8n0 e −2
√
n0 (x2 −x1 )
2
ẍ3 = −ωosc
x3 +
n0 (x2 −x1 )
3/2
√
− 8n0 e −2
√
3/2
8n0 e −2 n0 (x3 −x2 )
(9)
n0 (x3 −x2 )
(10)
(11)
Normal Modes of Coupled Oscillators
Gratuitous Video
Collision
The Lorenz Attractor
Normal Modes of Coupled Oscillators
The Lorenz Attractor
Dark Soliton Interactions: Normal Modes
Given the transcendental nature of equations (9)-(11) we cannot
directly apply the techniques used to find the normal modes for
coupled linear oscillators. In order to complete the analysis, we
must find a fixed point to the system of ODE’s, then linearize the
model about the fixed point. For Ω = 0.025, numerical package
was used to find the fixed point: (−4.3222, 0, 4.3222). Then the
linearized version of the ODE is given by the Jacobian matrix:


∂F1
∂F1
···
 ∂x1
∂xn 
 .
.. 
.

.
.
(12)
J= .
.
. 

 ∂Fm
∂Fm 
···
∂x1
∂xn
Normal Modes of Coupled Oscillators
The Lorenz Attractor
Dark Soliton Interactions: Normal Modes
Next, we evaluate the Jacobian at the fixed point and the analysis
of the normal modes proceeds as usual. The following eigenvalues
and eigenvectors were found.
n
eigenvalues
3
{−0.0088, −0.0031, −0.0003}
eigenvectors
 
   
1
1 
 1
−2 ,  0  , 1


1
−1
1
Normal Modes of Coupled Oscillators
Dark Soliton Interactions: Normal Modes
The Lorenz Attractor
Normal Modes of Coupled Oscillators
The Lorenz Attractor
The Lorenz Equations
Edward Lorenz was a researching climate models at MIT in the
late 1950’s. During this time, he became a pioneer of chaos theory
by inadvertently discovering a phenomena know as ”sensitive
dependence to initial conditions.”
ẋ = −σx + σy
(13)
ẏ = −xz + rx − y
(14)
ż = −xy − bz
(15)
Equations (13)-(14) are a reduced version of a model describing
Rayleigh-Bérnard convection. The parameter σ represents the
Prandtl number, while r is the Reynolds number.
Normal Modes of Coupled Oscillators
The Lorenz Attractor
The Lorenz Attractor
Figure: The Lorenz attractor with σ = 10, b = 8/3, r = 24.74.
Normal Modes of Coupled Oscillators
Separation of Initial Conditions
Figure: x10 = [10, 10, 10], x20 = [10.01, 10, 10].
The Lorenz Attractor
Normal Modes of Coupled Oscillators
The Lorenz Attractor
Volume Contraction in Phase Space
Consider V , a“blob” of initial conditions in phase space. It can be
show that the change in volume of that blob is given by the
following expression:
Z
∇ · fdV
V̇ =
(16)
V
Thus, for the Lorenz equations we have
∂
∂
∂
[σ(y − x)] +
[x(r − z) − y ] +
[xy − bz] (17)
∂x
∂y
∂z
= −σ − 1 − b < 0
(18)
∇·f =
Hence, the solution to (16) is V (t) = V (0)e −(σ+1+b)t . That is,
volumes contract exponentially fast in phase space. So, initial
conditions in V tend toward the same fixed points, limit cycles, or
strange attractor.
Normal Modes of Coupled Oscillators
The Lorenz Attractor
Bifurcations
Figure: One cycle solution with r = 325. Two cycle solution with
r = 225. Four cycle solution with r = 216.5.
Normal Modes of Coupled Oscillators
Transient Chaos: Phase Space
Figure: Bifurcations wrt the parameter r .
The Lorenz Attractor
Normal Modes of Coupled Oscillators
The Lorenz Attractor
Transient Chaos: Phase Space
Figure: Phase space solution exhibiting transient chaos with r = 22.
Normal Modes of Coupled Oscillators
The Lorenz Attractor
Transient Chaos: Time Series
Figure: Time series solution exhibiting transient chaos with r = 22.
Normal Modes of Coupled Oscillators
The Lorenz Attractor
Thank you.
”You’ve never heard of Chaos theory? Non-linear equations?
Strange attractors? Dr. Sattler, I refuse to believe you’re not
familiar with the concept of attraction.”
-Dr. Ian Malcom
Normal Modes of Coupled Oscillators
The Lorenz Attractor
Derivation of the Jacobian Matrix in 2D
Consider the system
ẋ = f (x, y )
(19)
ẏ = g (x, y )
(20)
and suppose (x ∗ , y ∗ ) is a fix point, i.e. f (x ∗ , y ∗ ) = g (x ∗ , y ∗ ) = 0.
Next, let u = x − x ∗ , v = y − y ∗ denote small perturbations away
from the fix point. Then, consider the u equation:
u̇ = ẋ
(21)
∗
∗
= f (x + u, y + v )
∂f ∂f ∗ ∗
+v
+ h.o.t.
= f (x , y ) + u
∂x (x ∗ ,y ∗ )
∂y (x ∗ ,y ∗ )
∂f ∂f +v
≈u
∂x (x ∗ ,y ∗ )
∂y (x ∗ ,y ∗ )
(22)
(23)
(24)
Normal Modes of Coupled Oscillators
The Lorenz Attractor
Derivation of the Jacobian Matrix in 2D
Similarly we find
v̇ ≈ u
∂g ∂g +
v
∂x (x ∗ ,y ∗ )
∂y (x ∗ ,y ∗ )
(25)
Thus we can write the linear matrix equation:
u̇
u
≈A
v̇
v
(26)
Where the Jacobian matrix A is given by
A=
∂f
∂x
∂g
∂x
∂f
∂y
∂g
∂y
!
(27)
(x ∗ ,y ∗ )