Chaos Theory

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Chaos Theory

• Ken Chan
• Supervisor Dr R C Johnson

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• 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
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• 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?
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• F exhibits sensitive dependence on initial
  conditions
• F is topologically transitive
• Periodic points are dense in the set

Chaos Theory
Conditions for chaos
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• 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
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• Construct a lattice on the real plane

Chaos Theory
Hyperbolic toral automorphisms
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• Construct a lattice on the real plane
• Identify the edges of the resulting square

Chaos Theory
Hyperbolic toral automorphisms
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Chaos Theory
Hyperbolic toral automorphisms
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Chaos Theory
Hyperbolic toral automorphisms
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• 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
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• 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
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• 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
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• 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
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• Observations
• Points quickly move apart.

Chaos Theory
Example The Cat Map
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• Observations
• Points quickly move apart.
• Eventually obtain original
• image
• - deterministic
• - not random!

Chaos Theory
Example The Cat Map
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• 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
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• Quantum Chaos
• Relativistic Chaos
• Fractals
• Bifurcations
• Time Series Analysis

Chaos Theory
Possible areas of research
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• 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
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• An Introduction to Dynamical Systems, Arrowsmith
  Place
• An Introduction to Chaotic Dynamical Systems,
  R.L.Devaney

Chaos Theory
Further information