1. Quasi-Periodicity & Chaos 1.QP & Poincare Sections 2.QP Route to Chaos 3.Universality in QP Route to Chaos 4.Frequency Locking 5.Winding Numbers 6.Circle Map 7.Devil’s Staircase & Farey Tree 8.Continued Fractions & Fibonacci Numbers 9.Chaos & Universality Revisited 10.Applications

2. Quasi-periodicity: Competition between 2 /more modes of incommensurate frequencies. Occurrences: Driven nonlinear oscillatory systems. Nonlinear systems with spontaneous creation of 2/more modes.

3. Actual measurements & calculations have finite precision: Results are all rational numbers. Can’t distinguish between quasi-periodicity & very long periodicity. Ditto chaos. Poincare map, divergence of nearby orbits helps but …. Caveat Emptor

4. Quasi-Periodicity & Poincare Sections 2-frequency dynamics: trajectories on surface of torus. Control freq =  R q points skips over p-1 pts (clockwise) Incommensurat e / long period: Drift ring Ex 6.2-1

5. Quasi-Periodicity Route to Chaos As parameter increases: One f.p. (if not driven ). Hopf bifurcation: f.p. → l.c. 2nd freq (torus T 2 ) → q.p. if incommensurate. 3rd freq (T 3 ) Small perturbation destroys torus → Chaos C.f., Landau scheme

6. Universality in QP Route to Chaos Reminder: Route to chaos theories only predict the possibility, not eventuality, of chaos. Quasi-periodicity route to chaos = Ruelle-Takens scenario (1971) Newhouse-Ruelle-Takens (78): 3- freq QP + perturbation → torus broken→ chaos with strange attractor Observe d 2 coupled nonlinear oscillators driven by sinusoidal signal: Weak coupling between oscillators: Parameter space: mostly 2- or 3-freq QP with occasional chaos. Strong coupling between oscillators: Parameter space: 2-freq QP regions mingle with chaotic ones. Reason: 3-freq QP easily destroyed by noises. 2 possible routes to chaos: QP or frequency locked. Universality: see §6.10.

7. Frequency Locking 2 modes are frequency-locked if they remain commensurate with the same ratio over a finite range of parameters. Moon + Earth, Mercury + Sun Reminders: Frequencies are amplitude dependent for non-linear oscillators. Frequency ratio remains constant in frequency-locking. Fourier analysis:  j = fundamental frequency of the j th oscillator Frequency-locking:  The (np) th harmonic of  1 resonates with the (nq) th harmonic of  2 for all positive integers n. FL → Resonance wins over amplitude dependence. Longer parameter ranges for FLs of small ratios. Benard

8. Winding Numbers Frequency-ratio parameter: at vanishing non- linearity & coupling Winding number:actual coupling Winding number = number of times the trajectory winds around the small cross-section of the torus after going once around the large circumference.  1 =  R ~ driving, circumference of torus,  2 =  r ~ characteristic, cross section of torus. For the Poincare section, integer parts of Ω & W are irrelevant. → Redefine them as:Ω modulo 1W modulo 1 W modulo 1 = fraction of circle trajectory travelled in 1 period of driving force.

9. Circle Map: Protype of QP Example: Poincare section: Δn = 1 → 1 period of driving force  m  Z

10. Winding number: ( no modulo 1 ) For the linear map Ω rational → periodic Ω irrational → quasi-periodic Increase ofΘin n units of time = Average frequency  2 ’ =

11. Sine-Circle Map Ω = bare winding number K = nonlinearity parameter = frequency ratio parameter  Map is not invertible for K > 1 → folding→ Chaos ? Fixed points: →

12. m = 0m = 1m = 2K = 0.5K = 2.0 Ω = 0.5ππ3π3πno f.p. Ω = 0.040.256.0312.3  1 f.p. Ω = 0.955.970.316.6  1 f.p. Values of 2π|Ω-m|: Stable if

13. Frequency-Locking →→ → m:1 frequency locking Stable fixed point for f: → Stable fixed point for f (k) : → → m:k frequency locking

14. K = 0.5, Ω = 0.04 1:2 locking K = 0.8, Ω = 0.5 f (2) 0:1 1:1 → m:k frequency locking K = 0.5, Ω = 0.94

15. Quasi-periodic K = 0.5, Ω = 0.6180339

16. Devil’s Staircase & Arnold’s Tongues W vs Ω for K = 1 Devil’s staircase Arnold’s tongues p:q locking: 0:1 locking: 1:1 locking:

17. The Farey Tree The rational fraction with the smallest denominator that lies between p/q and p’/q’ is (p+p’)/(q+q’) Range of Ω for lock p:q increases for decreasing q. (Mode resonance is larger for lower q ) Numerology approach: → Farley tree Between ¼ & ½ : 2/7,1/3,3/8,2/5,3/7

18. Consider 2 frequency lockings p/q and p’/q’ at respective parameters Ω and Ω’ that are adjusted so that their fixed points coincide. → By definition, all circle maps are “periodic modulo”. for any Θ and integer n (modulus action suspended) i.e. ∴ Adjust Ω’’ to get f is monotonic with Ω → Ω’’  [Ω, Ω’] Analytic Approach

19. Continued Fraction n th order approximation: Approximation of irrational number by rational fractions

20. for i = 1,2,3,… Golden Ratio →

21. Fibonacci numbers: 0, 1, 1, 2, 3, 5, 8, 13,…

22. A continued fraction a 0 +(a 1,a 2,…) is periodic with period k if there exists a M such that for all m > M Periodic fraction  solution to quadratic equation of integer coefficients Ex.6.9-2,3: Silver mean (a i = 2) = √2 - 1

23. Chaos & Universality Sine circle map ( global features ): K → 0 : quasi-periodic ( W irrational ) for most Ω. K → 0 : f q.p. = c ( 1-K ) β β ~ 0.34 (universal) K → 1 - : frequency-locked ( W rational ) for most Ω. K = 1 : f q.p. is fractal (universal). K > 1 : f non-invertible ~ chaos possible. frequency-locking & chaotic regions interwined. K ↑ beyond 1 for fixed f.l. Ω → period-doublings K ↑ beyond 1 for fixed q.p. Ω → chaos

24. Quasi-periodicity ( for K  1 ) : sequence of frequency lockings. e.g. Golden mean W via Fibonacci sequence { W n = F n /F n+1 }. f’(0) = 0 for K = 1Θ = 0 is part of the supercycles Define Ω p/q (K) as the freq ratio that gives rise to W = p/q Simplified notation for the case of the golden mean: is useless for calculations if W is irrational Sine circle map (local features) Ω n gives i.e.

25. d n (K) = distance from f(0) to nearest neighbor in the Ω n “supercycle”. Other forms of the scaling relations: Nab Golden meanself2.16443…0.52687… Silver meanself2.1748…0.5239…

26. Bifurcation Diagrams W kept at freq locking & increasing K pass 1 → periodic W kept at golden mean & increasing K pass 1 → Chaos Ω= 0.606661 Ω= 0.5

27. Universality Classes of Circle maps Criteria: functional form near inflection point (Θ = 0, K = 1). Prototype: z ~ degree of inflection z = 3 (cubic inflection) → δ,α same as sine circle map Large z: δ→ -4.11, α→ -1.0 Ref: B.Hu, A.Valinai, O.Piro, Phys.Lett., A 144, 7-10 (90)

28. Summary Smears in B.D. → Q.P. (K 1). For Ω rational, freq-locked tongue as K increases pass 1 (no immediate chaos). Above K = 1, Arnold tongues overlap → different starting θ’s lead to different freq-locking. Chaos can only be determined by Lyapunov exponents.

29. Applications Forced Rayleigh-Benard Convection Periodically Perturbed Cardiac Cells. Forced van der Pol Oscillator. K > 1 chaotic regimes in circle maps do not describe chaos in ODE systems (Torus not broken) K < 1: Freq-lock Q.P. → Chaos (2 parameters )

30. Forced Rayleigh-Benard Convection Ref: J.Stavans, et al, PRL 55, 596 (85)  R ~ convective flow  r ~ B = 200 G K ~ I 0 Mercury I 0 < 10 mA (circle-map-like): freq-lock, q.p., Arnold tongues … Golden mean seq → δ~ 2.8 Silver mean seq → δ~ 7.0 Fractal dim of q.p. regions: see chap 9.

31. Breaking up of the torus in a Benard experiment Ruelle-Takens route Bernard Experiment

32. Periodically Perturbed Cardiac Cells Glass & Mackey, McGill U. Periodic electrical simulations to culture of chick embryo heart cells Cell aggregates (from 7-day old embryonic chicks): d ~ 100μm. spontaneous beating: period ~ 0.5 s measured: time between beatings under simulation Simulation: reset phase Θ to Θ’ = g(Θ). Θ n = g(Θ n-1 ) + Ω(~ Sine circle map: f.l., q.p., A.T.,… ) q.p. chaotic

33. Comments on Biological Models Models: rough guides. Philosophical musings: pointless. Too much details → missing the point.

34. Van der Pol Oscillator Steady state chaos easier to observe if force is applied to Q . Experiment (op. amp): –>300 Arnold tongue observed. –Golden mean sequence: δ n (K) oscillates between -3.3 & -2.7 as n ↑. –Explanation: actual freq-lock & chaos are different transitions better described by “integrate & fire” model. Ref: P.Alstrom, et al, PRL 61, 1679 (88) Forced van der Pol Oscillator

35. P.Alstrom et al, PRL, 61, 1679 (88)