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Strange Attractors   From Art to Science J. C. Sprott Department of Physics University of Wisconsin - Madison Presented at...
Outline <ul><li>Modeling of chaotic data </li></ul><ul><li>Probability of chaos </li></ul><ul><li>Examples of strange attr...
Typical Experimental Data Time 0 500 x 5 -5
General 2-D Iterated Quadratic Map <ul><li>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 ...
Solutions Are Seldom Chaotic Chaotic Data (Lorenz equations) Solution of model equations Chaotic Data (Lorenz equations) S...
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 <ul><li>Limit set as  t       </li></ul><ul><li>Set of measure zero </li></ul><ul><li>Basin of attrac...
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) <ul><li>d x /d t = y </li></ul><ul><li>d y /d t  =  z </li></ul><ul><li>d z /d t  = - az  +  y 2  -  x </li>...
Linz and Sprott (1999) <ul><li>d x /d t = y </li></ul><ul><li>d y /d t  =  z </li></ul><ul><li>d z /d t  = - az  -  y  + |...
First Circuit
Bifurcation Diagram for First Circuit
Second Circuit
Third Circuit
Chaos Circuit
Summary <ul><li>Chaos is the exception at low  D </li></ul><ul><li>Chaos is the rule at high  D </li></ul><ul><li>Attracto...
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Strange Attractors

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Strange Attractors

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

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