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2. The Universality of Chaos

2. The Universality of Chaos. Some common features found in non-linear systems: Sequences of bifurcations (routes to chaos). Feigenbaum numbers. Ref: P.Cvitanovic, ” Universality in Chaos ” , 2nd ed.,Adam Hilger (89). 2.2. The Feigenbaum Numbers. See Appendix F. Logistic Map:

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2. The Universality of Chaos

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  1. 2. The Universality of Chaos • Some common features found in non-linear systems: • Sequences of bifurcations (routes to chaos). • Feigenbaum numbers. Ref: P.Cvitanovic,”Universality in Chaos”, 2nd ed.,Adam Hilger (89)

  2. 2.2. The Feigenbaum Numbers See Appendix F Logistic Map: Xn+1 = A xn (1-xn) Sine Map: Xn+1 = B sin(Πxn)

  3. Rate of convergence A1S 3.236 Size scaling:

  4. Numerical Determination of δ Value of A for the period 2n supercycle. Supercycles: Orbits that contain xmax

  5. Real Period-Doubling Systems Examples: diode circuit, fluid convection, modulated laser, acoustic waves, chemical reactions, mechanical oscillations, etc. Problem: δn measurable only for small n’s ( < 4 or 5 ) Lucky break: δ for logistic map converges very rapidly. Logistic-map-like region δ~ 3.57(10) δ~ 4.7(1)

  6. All disagreement are within 20%

  7. Traditional physics: Common behavior  common physical cause eg. Harmonic oscillations in low-excited systems → potential ~ quadratic around its minima Chaos / complexity : Common behavior  universality ( Common features in state space )

  8. 2.4. Using δ(see Chap 5) Quantitative predictions on system with unsolvable or unknown dynamical equations. Period –doubling systems: (if exists ) Ex 2.4-1, Show that

  9. Logistic map : Values taken from tables 2.1-2

  10. Diode experiment:

  11. 2.5. Feigenbaum Size Scaling Ratio of corresponding branches: Also: α and δ are about the right size for experimental observations

  12. 2.6. Self-similarity Fractals No inherent size-scale

  13. 2.8. Models & Universality Non-chaotic system : • Model: retain only relevant features. • Justification: prediction ~ observation. • Uniqueness: assumed. • Causality. Chaotic systems : • Universality → models not unique. • Common features ( not physical ) • No insight to microscopic structure gained. • Complexity

  14. 2.9. Computers & Chaos • Computers & graphics are crucial to study of chaos. • Divergence of nearby trajectories + runoff errors / noise → chaos • Question: Can any numerical computation be “meaningful” ? • Partial answer : Calculated result is always a possible evolution of the system, even though it may not be the one you wish to investigate. • Characteristics of system can still be studied in a statistical sense.

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