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Strange Attractors
 

Strange Attractors

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  • 05/27/09 Entire presentation available on WWW

Strange Attractors Strange Attractors Presentation Transcript

  • Strange Attractors From Art to Science J. C. Sprott Department of Physics University of Wisconsin - Madison Presented at the Santa Fe Institute On June 20, 2000
  • Outline
    • Modeling of chaotic data
    • Probability of chaos
    • Examples of strange attractors
    • Properties of strange attractors
    • Attractor dimension scaling
    • Lyapunov exponent scaling
    • Aesthetics
    • Simplest chaotic flows
    • New chaotic electrical circuits
  • Typical Experimental Data Time 0 500 x 5 -5
  • General 2-D Iterated Quadratic Map
    • x n +1 = a 1 + a 2 x n + a 3 x n 2 + a 4 x n y n + a 5 y n + a 6 y n 2
    • y n +1 = a 7 + a 8 x n + a 9 x n 2 + a 10 x n y n + a 11 y n + a 12 y n 2
  • Solutions Are Seldom Chaotic Chaotic Data (Lorenz equations) Solution of model equations Chaotic Data (Lorenz equations) Solution of model equations Time 0 200 x 20 -20
  • How common is chaos? Logistic Map x n +1 = Ax n (1 - x n ) -2 4 A Lyapunov Exponent 1 -1
  • A 2-D Example (Hénon Map) 2 b -2 a -4 1 x n +1 = 1 + ax n 2 + bx n -1
  • General 2-D Quadratic Map 100 % 10% 1% 0.1% Bounded solutions Chaotic solutions 0.1 1.0 10 a max
  • Probability of Chaotic Solutions Iterated maps Continuous flows (ODEs) 100% 10% 1% 0.1% 1 10 Dimension
  • Neural Net Architecture tanh
  • % Chaotic in Neural Networks
  • Types of Attractors Fixed Point Limit Cycle Torus Strange Attractor Spiral Radial
  • Strange Attractors
    • Limit set as t  
    • Set of measure zero
    • Basin of attraction
    • Fractal structure
      • non-integer dimension
      • self-similarity
      • infinite detail
    • Chaotic dynamics
      • sensitivity to initial conditions
      • topological transitivity
      • dense periodic orbits
    • Aesthetic appeal
  • Stretching and Folding
  • Correlation Dimension 5 0.5 1 10 System Dimension Correlation Dimension
  • Lyapunov Exponent 1 10 System Dimension Lyapunov Exponent 10 1 0.1 0.01
  • Aesthetic Evaluation
  • Sprott (1997)
    • d x /d t = y
    • d y /d t = z
    • d z /d t = - az + y 2 - x
    • 5 terms, 1 quadratic nonlinearity, 1 parameter
    “ Simplest Dissipative Chaotic Flow”
  • Linz and Sprott (1999)
    • d x /d t = y
    • d y /d t = z
    • d z /d t = - az - y + | x | - 1
    • 6 terms, 1 abs nonlinearity, 2 parameters (but one =1)
  • First Circuit
  • Bifurcation Diagram for First Circuit
  • Second Circuit
  • Third Circuit
  • Chaos Circuit
  • Summary
    • Chaos is the exception at low D
    • Chaos is the rule at high D
    • Attractor dimension ~ D 1/2
    • Lyapunov exponent decreases with increasing D
    • New simple chaotic flows have been discovered
    • New chaotic circuits have been developed