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. Logistic Map: X n+1 = A x n (1-x n ) Sine Map: X n+1 = B sin(Πx n ) 2.2. The Feigenbaum Numbers See Appendix F

3. Rate of convergence Size scaling: A1SA1S 3.236

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

5. Real Period-Doubling Systems Examples: diode circuit, fluid convection, modulated laser, acoustic waves, chemical reactions, mechanical oscillations, etc. Problem: δ n 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 α and δ are about the right size for experimental observations Ratio of corresponding branches: Also:

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.