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www.cse.ucsc.edu/classes/ams146/Spring05/index.html

Chapter 1A global overview of nonlinear dynamical systems and chaos

I.1 Definitions

• Dynamical system– A system of one or more variables which evolve in

time according to a given rule– Two types of dynamical systems:

• Differential equations: time is continuous

• Difference equations (iterated maps): time is discrete

I.1 Definitions

• Linear vs. nonlinear– A linear dynamical system is one in which

the rule governing the time-evolution of the system involves a linear combination of all the variables.

• EXAMPLE:

– A nonlinear dynamical system is simply… … not linear

I.1 Definitions

• Chaos: Poincaré: (1880) “ It so happens that small

differences in the initial state of the system can lead to very large differences in its final state. A small error in the former could then produce an enormous one in the latter. Prediction becomes impossible, and the system appears to behave randomly.”

I.1 Definitions

• Chaos: Poincaré: (1880) “ It so happens that small differences

in the initial state of the system can lead to very large differences in its final state. A small error in the former could then produce an enormous one in the latter. Prediction becomes impossible, and the system appears to behave randomly.”

I.1 Definitions

THE ESSENCE OF CHAOS!

• a dynamical system entirely determined by its initial conditions (deterministic)

• but which evolution cannot be predicted in the long-term

I.2 Examples of chaotic systems

• the solar system (Poincare)

• the weather (Lorenz)

• turbulence in fluids (Libchaber)

• solar activity (Parker)

• population growth (May)

• lots and lots of other systems…

I.2 Examples of chaotic systems

• the solar system (Poincare)

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I.2 Examples of chaotic systems

• the solar system (Poincare)

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I.2 Examples of chaotic systems

• the weather (Lorenz)

Difficulties in predicting the weather are not relatedto the complexity of the Earths’ climate but to CHAOS in the climate equations!

I.2 Examples of chaotic systems

• the weather (Lorenz)

I.2 Examples of chaotic systems

• turbulence in fluids

I.2 Examples of chaotic systems

• turbulence in fluids (Libchaber)

convection in liquid helium.

I.2 Examples of chaotic systems

• turbulence in fluids (similar experiment)

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I.2 Examples of chaotic systems

• turbulence in fluids (similar experiment)

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are needed to see this picture.

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I.2 Examples of chaotic systems

• turbulence in fluids - period doubling to chaos

I.2 Examples of chaotic systems

• solar activity

I.2 Examples of chaotic systems

• solar activity

QuickTime™ and aVideo decompressor

are needed to see this picture.

I.2 Examples of chaotic systems

• solar activity

I.2 Examples of chaotic systems

• solar activity

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

I.2 Examples of chaotic systems

• solar activity

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

I.2 Examples of chaotic systems

• solar activity

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

I.2 Examples of chaotic systems

• population growth (May) – discrete, apparently simple dynamical systems

can exhibit a rich array of behaviour– examples in nature are ubiquitous (population

dynamics, epidemic propagation, …)– discretization corresponds to breeding cycle,

seasonal recurrence, … – EXAMPLE: the logistic equation

Mathematical detour: cobweb diagrams for 1D maps

I.2 Examples of chaotic systems

• Logistic equation

r = 2 r = 3.3

I.2 Examples of chaotic systems

• Logistic equation

r = 3.5 r = 3.9

I.2 Examples of chaotic systems

• Logistic equation

I.3 Fractals

• Fractals appear everywhere in the study of dynamical systems

• Characteristics: – a fractal is a geometric object which can be

divided into parts, each of which is similar to the original object.

– they possess infinite detail, and are generally self-similar (independent of scale)

– they can often be constructed by repeating a simple process (a map, for instance) ad infinitum.

– they have a non-integer dimension!

I.3 Fractals

• A section through the Lorenz dynamical system yields a fractal somewhat like the Cantor set

an infinity of points, yet none are connected!

I.3 Fractals

• Bifurcation diagrams are also fractal

I.3 Fractals

• Bifurcation diagrams are also fractal

I.3 Fractals

• Also appear in nature.. What is the length of a coastline? (Mandelbrot)

I.3 Fractals

• Fractals naturally appear in iterated maps. – EXAMPLE: the Koch curve

I.3 Fractals

• Appear to be the fate of most 2D iterated maps– Example: The Mandelbrot set is built from

the quadratic map

where zn and c are complex numbers.

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