Spatiotemporal Chaos and Complexity

Chaos and Time-Series Analysis

12/12/00 Lecture #15 in Physics 505

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Homework #15 is due at 3:30 pm on 12/19 in my office or mailbox

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Note that HW #15 is graded differently from the others

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What topics did you enjoy most and least?
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How useful was it to have the lecture notes on the WWW?
Was the one long lecture per week the best format for the course?
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Comments on Homework #13 (Iterated Function Systems)

Everyone had good plots of Sierpinski triangle and fern
Should use small dots and many iterations
Use non-uniform probabilities (P ~ |det J| = |ad - bc|)
IFS patterns are attractors and fractals, but not usually called "strange attractors"
Bounded, linear IFS's are usually contracting in every direction, hence not chaotic
Only one person calculated capacity dimension (1.585)

Review (last week) - Non-Attracting Chaotic Sets

Multifractals

Fractals that are not uniformly covered
Spectrum of generalized dimensions, D_q
And generalized entropies, K_q
Where -infinity < q < +infinity

Summary of Time-Series Analysis
Time-Series Analysis Tutorial (using CDA)
Iterated Function Systems

Multiple affine transformations
Random iteration algorithm
Collage theorem
Image compression
IFS clumpiness test

Mandelbrot and Julia Sets

Non-Attracting Chaotic Sets

These sets ARE attracting
They are generally only transiently chaotic

Derivation from logistic equation

Start with logistic equation: X_n₊₁ = AX_n(1 - X_n)
Define a new variable: Z = A(1/2 - X)
Solve for X(Z, A) to get: X = 1/2 - Z/A
Substitute into logistic equation: Z_n₊₁ = Z_n² + c
Where c = A/2 - A²/4
Range (1 < A < 4) ==> -2 < c < 1/2
Z_n₊₁ = Z_n² + c is equivalent to logistic map
General the above to complex values of Z and c

Review of complex numbers

Z = X + iY, where i = (-1)^1/2
Z² = X² + 2iXY - Y²
Separate real and imaginary parts

X_n₊₁ = X_n² - Y_n² + a
Y_n₊₁ = 2X_nY_n + b
where a = Re(c) and b = Im(c)

This is just another 2-D quadratic map
X, Y, a, and b are real variables
Orbits are either bounded or unbounded

Mandelbrot (M) set

Region of a-b space with bounded orbits with X₀ = Y₀ = 0
Orbit escapes to infinity if X² + Y² > 4 (circle of radius 2)
It's sometimes defined as the complement of this
There is only one Mandelbrot set
The "buds" in the M-set correspond to different periodicities
Usually plotted are escape-time contours in colors
Each point in the M-set has a corresponding Julia set
The M-set is everywhere connected
Boundary of M-set is fractal with dimension = 2 (proved)
Area of set is ~ p/2
Points along the real axis replicate logistic map and exhibit chaos
Points just outside the boundary exhibit transient chaos
The chaotic region appears to be a set of measure zero (not proved)
Boundary of M-set is a repellor
With deep zoom, M-set and J-set are identical
People have zoomed in by factors as large as 10¹⁶⁰⁰
Miniature M-sets are found at deep zooms
See the Mandelbrot Java applet written by Andrew R. Cavender

Julia (J) sets

Region of X₀-Y₀ space with bounded orbits for given a, b
Orbit escapes to infinity if X² + Y² > 4 (circle of radius 2)
This is sometimes called the "filled-in" Julia set
There are infinitely many J-sets
Usually plotted are escape-time contours in colors
The J-sets correspond to points on the Mandelbrot set
J-sets from inside the M-set are connected
J-sets from outside the M-set are "dusts"
Boundary of J-set is a repellor
With deep zoom, J-set and M-set are identical

Fixed points of Julia sets

Z = Z² + c ==> Z = 1/2 ± (1 - 4c)^1/2/2
These fixed points are unstable (repellors)
They can be found by backward iteration: Z_n = ± (Z_n₊₁ - c)^1/2
There are two roots (pre-images) each with two roots, etc.
Find them with the random iteration algorithm (cf: IFS)
The repelling boundary of J-set thus becomes an attractor
An example is the Julia dendrite (c = i)

Generalized Julia sets

Other complex functions Z_n₊₁ = F(Z_n) have interesting boundaries
No good answer to what's special about these functions
General 2-D quadratic map sometimes have interesting basin boundaries
Fractal eXtreme has a plug-in for calculating these

Applications of M-set and J-sets

None known except computer art
High traction shoe tread?

Spatiotemporal Chaos (Complexity)

Examples of spatiotemporal (infinite-dimensional) dynamics:

Turbulent fluids
Plasmas (ionized gases)
The weather
Molecular diffusion
The brain
Any process governed by partial differential equations (PDEs)

Two coupled logistic maps

X_n₊₁ = (1 - e) A₁X_n (1 - X_n) + eA₂ Y_n (1 - Y_n)

Y_n

₊₁

₂

Y_n

₁

X_n

Each map has a different A but the same coupling 0 < e < 1
If one map is periodic and one chaotic, which wins?
Can do the same with other maps and flows (Lorenz, etc.)

Coupled-map lattices (CMLs)

Consider a 1-D lattice of logistic maps
Coupling can be to all others (GCMs) or N neighbors
Dynamics exhibit transient chaos, waves, diffusion, damping, etc.
Such systems have not been extensively studied
Could use a distribution of e values
Could extend calculations to 2 or more dimensions

Cellular automata

1-D example with periodic boundary conditions
2-D example (Game of Life)
Many other CA rules and models are possible
Local rules give rise to global behavior
Visit the Primordial Soup Kitchen (Prof. David Griffeath)

Langton's ants

Example of simple computer automaton
Lots of ways to extend these models

Summary of spatiotemporal (complex) models

Models can be discrete or continuous in space, time, and value:

space D, time D, value D	cellular automata
space D, time D, value C	coupled map lattices
space D, time C, value D	not studied
space D, time C, value C	coupled flow lattices
space C, time D, value D	not studied
space C, time D, value C	not studied
space C, time C, value D	not studied
space C, time C, value C	partial differential eqns

Concluding Remarks

Nature is nonlinear and often chaotic
Nature is complex, but simple models may suffice
These models preclude prediction but invite control
Remember the butterfly!

Please contact me if you want to do additional work in this area.
Best wishes!

J. C. Sprott | Physics 505 Home Page | Previous Lecture | Next Lecture