Strange Attractors
Chaos and TimeSeries Analysis
10/10/00 Lecture #6 in Physics 505
Comments on Homework
#4 (Lorenz Attractor)

Everyone did a good job

To get smooth graphs, make h smaller or connect the dots
Review (last
week)  Lyapunov Exponents

Lyapunov Exponents are a dynamical measure of chaos

There are as many exponents as the system has dimensions

dV/dt / V = l_{1}
+ l_{2} + l_{3}
+ ...

= <log det J> for maps

= <trace J> = <f_{x} + g_{y}
+ h_{z} + ...> for flows

Where J is the Jacobian matrix

Sl must be negative for an attractor

Sl must be zero for a conservative
(Hamiltonian) system

For chaos we require l_{1}
> 0 (at least one positive LE)

For 1D Maps, l = <log df/dX>

2D example, Hénon
map:

X_{n}_{+1} = 1  CX_{n}^{2}
+ BY_{n} [= f(X, Y)]

Y_{n}_{+1} = X_{n}
[= g(X, Y)]

Usual parameters for chaos: B = 0.3, C = 1.4

l_{1} + l_{2}
= <log f_{x}g_{y}  f_{y}g_{x}>
= log B = 1.204 (basee)

Numerical calculation gives l_{1}
= 0.419 (basee)

Hence l_{2} = 1.204  0.419 = 1.623
(basee)

Fixed points at x* = y*
= 1.1313445 and x* = y* = 0.63133545

General character of Lyapunov exponents in flows:

(, , , , ...) fixed point (0D)

(0, , , , ...) limit cycle (1D)

(0,0, , , ...) 2torus (2D)

(0, 0, 0, , ...) 3torus, etc. (3D, etc.)

(+, 0, , , ...) strange (chaotic) (2^{+}D)

(+, +, 0, , ...) hyperchaos, etc. (3^{+}D)

Numerical Calculation of Largest Lyapunov
Exponent

Start with any initial condition in the basin of attraction

Iterate until the orbit is on the attractor

Select (almost any) nearby point (separated by d_{0})

Advance both orbits one iteration and calculate new separation d_{1}

Evaluate log d_{1}/d_{0} in any convenient
base

Readjust one orbit so its separation is d_{0} in same
direction as d_{1}

Repeat steps 46 many times and calculate average of step 5

The largest Lyapunov exponent is l_{1}
= <log d_{1}/d_{0}>

If map approximates an ODE, then l_{1}
= <log d_{1}/d_{0}> / h

A positive value of l_{1} indicates
chaos

Shadowing lemma: The computed orbit shadows some possible
orbit

KaplanYorke (Lyapunov) Dimension

Attractor dimension is a geometrical measure of complexity

Random noise is infinite dimensional (infinitely complex)

How do we calculate the dimension of an attractor? (many ways)

Suppose system has dimension N (hence N Lyapunov exponents)

Suppose the first D of these sum to zero

Then the attractor would have dimension D

(in D dimensions there would be neither expansion nor contraction)

In general, find the largest D for which l_{1}
+ l_{2} + ... + l_{D}
> 0

(The integer D is sometimes called the topological dimension)

The attractor dimension would be between D and D + 1

However, we can do better by interpolating:

D_{KY} = D + (l_{1}
+ l_{2} + ... + l_{D})
/ l_{D+1}

The KaplanYorke conjecture is that D_{KY} agrees
with other methods

2D Map Example: Hénon map
(B = 0.3, C = 1.4)

l_{1} = 0.419 and l_{2}
= 1.623

D = 1 and D_{KY} = 1 + l_{1}
/ l_{2} = 1 + 0.419 / 1.623 = 1.258

Agrees with intuition and other calculations

3D Flow Example: Lorenz Attractor (p = 10, r
= 28, b = 8/3)

Numerical calculation gives l_{1}
= 0.906

Since it is a flow, l_{2} = 0

l_{1} + l_{2}
+ l_{3} = <f_{x} +
g_{y}
+ h_{z}> = p  1  b = 13.667

Therefore, l_{3} = 14.572

D = 2 and D_{KY} = 2 + l_{1}
/ l_{3} = 2 + 0.906 / 14.572 =
2.062

Chaotic flows always have D_{KY} > 2

Chaotic maps always have D_{KY} > 1

Higher order interpolations are possible

Precautions

Be sure orbit is bounded and looks chaotic

Be sure orbit has adequately sampled the attractor

Watch for contraction to zero within machine precision

Test with different initial conditions, step size, etc.

Supplement with other tests (Poincaré section, Power spectrum,
etc.)
J. C. Sprott  Physics 505
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