Dynamical Systems Theory
Chaos and TimeSeries Analysis
9/26/00 Lecture #4 in Physics 505
Comments on Homework #2 (Bifurcation Diagrams)
Most everyone did fine
Number of periodic windows increases with period
1 period3
2 period4
3 period5
5 period6, etc.
Scaling of number and window width with period is an open question
Review (
last week
)  Nonchaotic Multidimensional Flows
Maps
versus
flows
(
discrete
versus
continuous
time)
Exponential growth or decay (1D linear)
Equation:
d
x
/d
t
=
ax
Solution:
x
=
x
_{0}
e
^{at}
Logistic differential equation (1D nonlinear)
Equation:
d
x
/d
t
=
ax
(1 
x)
Equilibrium points
at
x*
= 0 and
x*
= 1
Stability
determined by sign of d
f
/d
x
Stable equilibrium point is approached
asymptotically
Oscillations and chaos are not possible
Circular motion (2D linear)
Equations:
d
x
/d
t
=
y
, d
y
/d
t
= 
x
Solution:
x
=
x
_{0 }
cos
t
,
y
= 
x
_{0}
sin
t
Solutions are circles around a
center
at (0, 0)
Center is
neutrally stable
(neither attracts nor repels)
Mass on a spring (2D linear)
Equations:
d
x
/d
t
=
v
, d
v
/d
t
= 
x
Same as circular motion above
(
x
,
v
) are
phasespace
variables
Trajectory forms a
phase portrait
Mass on a spring with friction (dissipative 2D linear)
Equations:
d
x
/d
t
=
v
, d
v
/d
t
= 
x

bv
Solution attracts to a stable
equilibrium point
at (0, 0)
Nonautonomous
systems
These systems have an explicit
time dependence
Can remove time by defining a
new variable
Dimension increases
by 1 (circle becomes a
torus
)
Limit cycles
(van der Pol equation)
Numerical methods
for solving ODEs
Suppose d
x
/d
t
=
f
(
x
,
y
) and d
y
/d
t
=
g
(
x
,
y
)
Let
h
be a small increment of time
Euler
method
Leapfrog
method
Second order
RungeKutta
Fourth order
RungeKutta
General 2D Linear Flows
Equilibrium
:
x*
= 0,
y
* = 0 (stable)
Mechanical energy is not conserved (
dissipation
)
Best undersood from
phase portrait
Types of
equilibrium points
in linear 2D systems:
Spiral point
(focus)
if
b
< 2 (underdamped)
Radial point
(node)
if
b
> 2 (overdamped)
Saddle point
(hyperbolic point)
Attractor >
sink
, repellor >
source
Trajectory
cannot intersect
itself
PoincareBendixson
theorem (no chaos in 2D)
Chaos in flows requires at least
3 variables
and a
nonlinearity
Other limit cycle examples (2D nonlinear)
Circular limit cycle
d
x
/d
t
=
y
d
y
/d
t
= (1 
x
^{2}

y
^{2}
)
y

x
Unstable
equilibrium point
:
x
* = 0:
y
* = 0
Solution:
x
^{2}
+
y
^{2}
= 1 (a circle)
Bad
numerical method
(Euler) can give
chaos
Lorenz example (in
The Essence of Chaos
)
d
x
/d
t
=
x

y

x
^{3}
d
y
/d
t
=
x

x
^{2}
y
Unstable
equilibrium point
:
x
* = 0:
y
* = 0
Limit cycle is approximately a circle of radius 1
Bad
numerical method
(Euler) can give
chaos
Stability of Equilibria in 2 Dimensions
Recall
1D result
:
d
x
/d
t
=
f
(
x
)
Equilibrium
:
f
(
x
) = 0 ==>
x
*
Stability
is determined by sign of d
f
/d
x
at
x
*
Solution
near equilibrium is
x
=
x
* + (
x
_{0}

x
*)e
^{lt}
Where
l
= d
f
/d
x
(at
x
=
x
*) is the
growth rate
d
x
/d
t
=
f
(
x
,
y
)
d
y
/d
t
=
g
(
x
,
y
)
Equilibrium points
:
f
=
g
= 0 ==>
x
*,
y
*
Calculate the
Jacobian
matrix
J
at
x
*,
y
*
Let
f
_{x}
be the partial derivative of
f
with respect to
x
, etc.
The
eigenvalues
of
J
are: (
f
_{x}

l
)(
g
_{y}

l
) =
f
_{y}
g
_{x}
This is the
characteristic equation
(quadratic in 2D, etc.)
Solutions
are of the form:
x
=
x
_{0}
e
^{lt}
,
y
=
y
_{0}
e
^{lt}
There are
2 solutions
for
l
(real or complex conjugates)
Example: Damped Harmonic Oscillator (2D linear)
d
x
/d
t
=
y
(velocity)
d
y
/d
t
= 
x

by
(force)
f
_{x}
= 0,
f
_{y}
= 1,
g
_{x}
= 1,
g
_{y}
= 
b
Characteristic equation
:
l
^{2}
+
b
l
+ 1 = 0
Solutions
:
l
= 
b
/2 ± (
b
^{2}
 4)
^{1/2}
/2 (
eigenvalues
)
First case:
b
> 2
Overdamped
Two negative real eigenvalues
This gives a radial point (node)
Second case:
b
= 2
Critical damping
Two negative equal eigenvalues
Third case: 0 <
b
< 2
Underdamped
Two negative complex eigenvalues
This gives a spiral point (focus)
Fourth case:
b
< 0 (negative damping)
Exponential growth
Two positive eigenvalues
Attractors become repellors
Saddle Points (or hyperbolic points)
Example
(note similarity to harmonic oscillator)
d
x
/d
t
=
y
d
y
/d
t
= x
Eigenvalues
are
l
_{1}
= 1,
l
_{2}
= 1
The flow is of the form:
Unstable manifold
(outset)
l
_{1}
> 0
Stable manifold
(inset)
l
_{2}
< 0
Also called
separatrices
(trajectories can't cross)
Separatrices given by the
eigenvectors
of
J
R
=
l
R
Consult any book on
linear algebra
We
won't be using
the eigenvectors
The separatrices
organize the phase space
The eigenvalues allow
prediction of bifurcations
Area Contraction (or expansion) in 2D
l
_{1}
l
_{2}
= det
J
=
f
_{x}
g
_{y}

f
_{y}
g
_{x}
(
determinant
of
J
)
l
_{1}
+
l
_{2}
= trace
J
=
f
_{x}
+
g
_{y}
(
trace
of
J
)
Expanding
direction: d
E
/d
t
=
l
_{1}
E
(
l
_{1}
> 0)
Contracting
direction: d
C
/d
t
=
l
_{2}
C
(
l
_{2}
< 0)
Phase space area
:
A
=
CE
sin
q
d
A
/d
t
=
C
d
E
/d
t
sin
q
+
E
d
C
/d
t
sin
q
=
CE
(
l
_{1}
+
l
_{2}
) sin
q
d
A
/d
t
/
A
=
l
_{1}
+
l
_{2}
(fractional rate of expansion)
In higher dimension:
d
V
/d
t
/
V
=
l
_{1}
+
l
_{2}
+ ...
(sum of eigenvalues)
V
is the
phasespace volume
of initial conditions
Sum of the eigenvalues must be negative for an attractor
Flows in 3 Dimensions
Types of equilibria
Cubic
characteristic equation
Three eigenvalues
(3 real or 1 real)
Attracting
equilibrium points (2 types)
Repelling
equilibrium points (2 types)
Saddle points
(4 types)
Index
: number of eigenvalues with Re(
l
) > 0
[or dimension of the unstable manifold]
Chaos
occurs with 2 or more unstable equilibria
Attractors
in 3D flows
Equilibrium point
(as in 1 and 2 dimensions)
Limit cycle
(as in 2 dimensions)
Torus
(
quasiperiodic
 2
incommensurate
frequencies)
Strange
(chaotic) attractors
Examples of
chaotic dissipative flows
in 3D:
Driven pendulum
d
x
/d
t
=
v
d
v
/d
t
= sin
x

bv
+
A
sin
wt
A
= 0.6,
b
= 0.05,
w
= 0.7
Driven nonlinear oscillator (Ueda)
d
x
/d
t
=
v
d
v
/d
t
= 
x
^{3}

bv
+
A
sin
wt
A
= 2.5,
b
= 0.05,
w
= 0.7
Driven Duffing oscillator
d
x
/d
t
=
v
d
v
/d
t
=
x

x
^{3}

bv
+
A
sin
wt
A
= 0.7,
b
= 0.05,
w
= 0.7
Driven Van der Pol oscillator
d
x
/d
t
=
v
d
v
/d
t
= 
x
+
b
(1 
x
^{2}
)
v
+
A
sin
wt
A
= 0.61,
b
= 1,
w
= 1.1 (
a torus
)
Lorenz attractor
d
x
/d
t
=
p
(
y

x
)
d
y
/d
t
= 
xz
+
rx

y
dz/dt =
xy

bz
p
= 10,
r
= 28,
b
= 8/3
Rössler attractor
d
x
/d
t
= 
y

z
d
y
/d
t
=
x
+
ay
d
z
/d
t
=
b
+
z
(
x

c
)
a
=
b
= 0.2,
c
= 5.7
Simplest dissipative chaotic flow
d
x
/d
t
=
y
d
y
/d
t
=
z
d
z
/d
t
= 
x
+
y
^{2}

Az
A
= 2.107
Other simple chaotic flows
J. C. Sprott

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