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 ID: 570081
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Slide1
Introduction to Chaos
by: Saeed
Heidary
29 Feb 2013Slide2
Outline:
Chaos in Deterministic Dynamical systems
Sensitivity to initial conditions
Lyapunov
exponent
Fractal geometry
Chaotic time series predictionSlide3
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.Slide4
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 predictionSlide5
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.Slide6
L
yapunov Exponent
Tow near points in phase space diverge exponentiallySlide7
Lyapunov
exponent
Stochastic (random ) systems:
Chaotic systems :
Regular systems : Slide8
Chaos and Randomness
Chaos is NOT randomness though it can look pretty random.
Let us have a look at two time series:Slide9
Chaos and Randomness
x
n+
1
=
1.4
- x
2
n
+
0.3
y
n
y
n+1 = xnWhite NoiseNon - deterministicHenon MapDeterministicplot xn+1 versus xn (phase space)Slide10
fractals
Geometrical objects generally with
non-integer
dimension
Self-similarity (contains infinite copies of itself)
Structure on all scales (detail persists when zoomed arbitrarily)Slide11
Fractals production
Applying simple rule against simple shape and
iterate
itSlide12
Fractal productionSlide13
Sierpinsky
carpetSlide14
Broccoli fractal!Slide15
Box counting dimensionSlide16
Integer dimension
Point 0
Line 1
Surface 2
Volume 3Slide17
Exercise for non-integer dimension
Calculate box counting dimension for cantor set and repeat it for
sierpinsky
carpet?Slide18
Fractals in natureSlide19
Fractals in natureSlide20
Complexity - disorder
Nature is complicated
but
Simple models may suffice
I emphasize:
“Complexity doesn’t mean disorder.”Slide21
Prediction in chaotic time series
Consider 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?)Slide22
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 Slide23
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:Slide24
Local Linear Approximation (LLA) Method
Again
the unknown coefficients can be solved using a least – squares method
Finally we have prediction Slide25
THANK YOU