Introduction to Chaos

by: Saeed Heidary

29 Feb 2013

Outline:Chaos in Deterministic Dynamical systems

Sensitivity to initial conditions

Lyapunov exponent

Fractal geometry

Chaotic time series prediction

Chaos in Deterministic Dynamical systems

There are not any random terms in the equation(s) which describe evolution of the deterministic system.

If the these equations have non-linear term,the

system may be chaotic .

Nonlinearity is a necessary condition but not enough.

Characteristics of chaotic systems

Sensitivity to initial conditions(butterfly effect)

Sensitivity measured by lyapunov exponent.

complex shape in phase space (Fractals ) Fractals are shape with fractional (non integer) dimension !.

Allow short-term prediction but not long-term prediction

Butterfly Effect

The butterfly effect describes the notion that the flapping of the wings of a butterfly can ‘cause’ a typhoon at the other side of the world.

Lyapunov ExponentTow near points in phase space diverge exponentially

Lyapunov exponent

Stochastic (random ) systems:

Chaotic systems :

Regular systems :

0

0

Chaos and RandomnessChaos is NOT randomness though it can look pretty random.

Let us have a look at two time series:

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350 400-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350 400

Chaos and Randomness

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350 400

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350 400

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

xn+1 = 1.4 - x2n + 0.3 yn

yn+1 = xn

White NoiseNon - deterministic

Henon MapDeterministic

plot xn+1 versus xn (phase space)

fractalsGeometrical objects generally with non-integer

dimensionSelf-similarity (contains infinite copies of itself)Structure on all scales (detail persists when zoomed

arbitrarily)

Fractals productionApplying simple rule against simple shape and iterate it

Fractal production

Sierpinsky carpet

Broccoli fractal!

Box counting dimension

Integer dimensionPoint 0 Line 1

Surface 2

Volume 3

Exercise for non-integer dimensionCalculate box counting dimension for cantor set and

repeat it for sierpinsky carpet?

Fractals in nature

Fractals in nature

Complexity - disorder

Nature is complicated

but

Simple models may suffice

I emphasize:

“Complexity doesn’t mean disorder.”

Prediction in chaotic time seriesConsider a time serie :

The goal is to predict

T is small and in the worth case is equal to inverse of lyapunov exponent of the system (why?)

Forecasting chaotic time series procedure (Local Linear Approximation)The first step is to embed the time series to obtain the

reconstruction (classify)

The next step is to measure the separation distance between the vector and the other reconstructed vectors

And sort them from smalest to largest

The (or ) are ordered with respect to

Local Linear Approximation (LLA) Method

the next step is to map the nearest neighbors of forward forward in the reconstructed phase space for a time T

These evolved points are The components of these vectores are as follows:

Local linear approximation:

Local Linear Approximation (LLA) Method

Again

the unknown coefficients can be solved using a least – squares method

Finally we have prediction

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