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Chaos Theory

- Ken Chan
- Supervisor Dr R C Johnson

- What is Chaos Theory?
- Conditions for chaos
- A chaotic system hyperbolic toral automorphisms
- An example the cat map
- Areas of research
- Applications

Chaos Theory

Introduction

- Nonlinear systems
- Highly sensitive dependence on initial

conditions - Not complete disorder!
- Examples The weather
- The stock market
- Population growth models

Chaos Theory

What is chaos theory?

- F exhibits sensitive dependence on initial

conditions - F is topologically transitive
- Periodic points are dense in the set

Chaos Theory

Conditions for chaos

- F exhibits sensitive dependence on initial

conditions - small variations of initial conditions may

produce large variations in the long term - F is topologically transitive
- cannot be decomposed into two disjoint open sets
- Periodic points are dense in the set
- A is dense in X if the closure of A is X

Chaos Theory

Conditions for chaos

- Construct a lattice on the real plane

Chaos Theory

Hyperbolic toral automorphisms

- Construct a lattice on the real plane
- Identify the edges of the resulting square

Chaos Theory

Hyperbolic toral automorphisms

Chaos Theory

Hyperbolic toral automorphisms

Chaos Theory

Hyperbolic toral automorphisms

- Consider a periodic point in T
- Unstable set is dense in T
- Sensitive dependence on the entire torus

Chaos Theory

Sensitive dependence on initial conditions

- Consider a periodic point p1, p2 in T
- Write as
- Union of all such point is the entire space

Chaos Theory

Periodic points are dense in the torus

- any point can eventually be mapped to any other

point - Idea 1. Consider two sets U, V.
- 2. Points p in U, q in V
- 3. Find k such that Fk(p) is in V

Chaos Theory

Topological transitivity

- Matrix
- -symmetric
- -hyperbolic
- Perpendicular action because of symmetric matrix
- Stretching and folding feature of chaotic

systems - Note Area preserving

Chaos Theory

Example The Cat Map

- Observations
- Points quickly move apart.

Chaos Theory

Example The Cat Map

- Observations
- Points quickly move apart.
- Eventually obtain original
- image
- - deterministic
- - not random!

Chaos Theory

Example The Cat Map

- Chaos Theory is the study of certain nonlinear

systems which - exhibit sensitive dependence on initial

conditions - Conditions 1. Periodic points are dense in the

set - 2. Topologically transitive
- 3. Sensitive dependence on initial conditions
- Chaotic systems are deterministic and not random
- Stretching and folding is a common feature of

chaotic systems - Example

Chaos Theory

Summary

- Quantum Chaos
- Relativistic Chaos
- Fractals
- Bifurcations
- Time Series Analysis

Chaos Theory

Possible areas of research

- Weather forecast
- Population growth models
- Satellites
- Chemical reactions
- The solar system
- The flow of blood in the body
- Effects of turbulence
- And more

Chaos Theory

Applications

- An Introduction to Dynamical Systems, Arrowsmith

Place - An Introduction to Chaotic Dynamical Systems,

R.L.Devaney

Chaos Theory

Further information

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