Nonlinear Physics

1. Nonlinear Physics • Textbook: • R.C.Hilborn, “Chaos & Nonlinear Dynamics”, 2nd ed., Oxford Univ Press (94,00) • References: • R.H.Enns, G.C.McGuire, “Nonlinear Physics with Mathematica for Scientists & Engineers”, Birhauser (01) • H.G.Schuster, “Deterministic Chaos”, Physik-Verlag (84) • Extra Readings: • I.Prigogine, “Order from Chaos”, Bantam (84) Website: http://ckw.phys.ncku.edu.tw (shuts down on Sundays) Home work submission: class@ckw.phys.ncku.edu.tw

2. Linear & Nonlinear Systems • Linear System: • Equation of motion is linear. X’’ + ω2x = 0 • linear superposition holds: f, g solutions → αf + βg solution • Response is linear • Nonlinear System: • Equation of motion is not linear. X’’ + ω2x2 = 0 • Projection of a linear equation is often nonlinear. • Linear Liouville eq → Nonlinear thermodynamics • Linear Schrodinger eq → Quantum chaos ? • Sudden change of behavior as parameter changes continuously, cf., 2nd order phase transition.

3. Two Main Branches of Nonlinear Physics: • Chaos • Solitons • What is chaos ? • Unpredictable behavior of a simple, deterministic system • Necessary Conditions of Chaotic behavior • Equations of motion are nonlinear with DOF  3. • Certain parameter is greater than a critical value. • Why study chaos ? • Ubiquity • Universality • Relation with Complexity

4. Plan of Study • Examples of Chaotic Sytems. • Universality of Chaos. • State spaces • Fixed points analysis • Poincare section • Bifurcation • Routes to Chaos • Iterated Maps • Quasi-periodicity • Intermittency & Crises • Hamiltonian Systems

5. Ubiquity • Some Systems known to exhibit chaos: • Mechanical Oscillators • Electrical Cicuits • Lasers • Optical Systems • Chemical Reactions • Nerve Cells, Heart Cells, … • Heated Fluid • Josephson Junctions (Superconductor) • 3-Body Problem • Particle Accelerators • Nonlinear waves in Plasma • Quantum Chaos ?

6. Three Chaotic Systems • Diode Circuit • Population Growth • Lorenz Model R.H.Enns, G.C.McGuire, “Nonlinear Physics with Mathematica for Scientists & Engineers”, Birhauser (01)

7. Specification of a Deterministic Dynamical System • Time-evolution eqs ( eqs of motion ) • Values of parameters. • Initial conditions. Deterministic Chaos

8. Questions • Criteria for chaos ? • Transition to chaos ? • Quantification of chaos ? • Universality of chaos ? • Classification of chaos ? • Applications ? • Philosophy ?

9. Diode Circuit R.W.Rollins, E.R.Hunt, Phys. Rev. Lett. 49, 1295 (82) • Becomes capacitor when reverse biased. • Becomes voltage source -Vd = Vf when forward biased.

10. Cause of bifurcation: After a large forward bias current Im , the diode will remain conducting for time τr after bias is reversed, i.e., there’s current flowing in the reverse bias direction so that the diode voltage is lower than usual. Reverse recovery time =

11. Bifurcation

12. period 4 period 8 Period 4

13. Divergence of evolution in chaotic regime

14. I(t) sampled at period of V(t) Period 4 Bifurcation diagram

15. Larger signal In Period 3 in window

16. Summary • Sudden change ( bifurcation ) as parameter ( V0 ) changes continuously. • Changes ( periodic → choatic ) reproducible. • Evolution seemingly unrelated to external forces. • Chaos is distinguishable from noises by its divergence of nearby trajectories.

17. Population Growth R.M.May M.Feigenbaum Logistic eq. Iterated Map Iteration function

18. Maximum: Fixed point → if if

19. A = 0.9

20. X0=0.1 X0=0.8 A = 1.5

21. N=5000 A = 1.0

22. A = 3.1

24. Poincare section 1-D iterated map ~ 3-D state space Dimension of state space = number of 1st order autonomous differential eqs. Autonomous = Not explicitly dependent on the independent variable. Diode circuit is 3-D.

25. Lorenz Model

26. Derivation of the Lorenz eqs.: Appendix C Navier-Stokes eqs. + Entropy Balance eq. L.E.Reichl, ”A Modern Course in Statistical Physics”, 2nd ed., §10.B, Wiley (98). X ~ ψ(t) Stream function (fluid flow) Y ~ T between ↑↓ fluid within cell. Z ~ Tfrom linear variation as function of z. r < rC : conduction r > rC : convection

27. Dynamic Phenomena found in Lorenz Model • Stable & unstable fixed points. • Attractors (periodic). • Strange attractors (aperiodic). • Homoclinic orbits (embedded in 2-D manifold ). • Heteroclinic orbits ( connecting unstable fixed point & limit cycle ). • Intermittency (almost periodic, bursts of chaos) • Bistability. • Hysteresis. • Coexistence of stable limit cycles & chaotic regions. • Various cascading bifurcations.

28. 3 fixed points at (0,0,0) & repulsive r < 1 attractive r = 1 is bifurcation point attractive r > 1 repulsive repulsive regions outside atractive ones, complicated behavior. r > 14 repulsive r = 160 : periodic. X oscillates around 0 → fluid convecting clockwise, then anti-clockwise, … r = 150 : period 2. r = 146 : period 4. … r < 144 : chaos

29. Back

30. Back

31. Back

32. Intermittence Back

33. Period 1 Back

34. Period 2 Back

35. Period 4 Back

36. Chaos

37. Divergence of nearby orbits

38. Determinism vs Butterfly Effect • Divergence of nearby trajectories → Chaos → Unpredictability • Butterfly Effect • Unpredictability ~ Lack of solution in closed form • Worst case: attractors with riddled basins. • Laplace: God = Calculating super-intelligent → determinism (no free will). • Quantum mechanics: Prediction probabilistic. Multiverse? Free will? • Unpredictability: Free will?