Quasi-Periodicity & Chaos

1. QP & Poincare Sections2. QP Route to Chaos3. Universality in QP Route to Chaos4. Frequency Locking5. Winding Numbers6. Circle Map7. Devil’s Staircase & Farey Tree8. Continued Fractions & Fibonacci

Numbers9. Chaos & Universality Revisited10.Applications

Quasi-periodicity:

Competition between 2 /more modes of incommensurate frequencies.

Occurrences:

• Driven nonlinear oscillatory systems.

• Nonlinear systems with spontaneous creation of 2/more modes.

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

Quasi-Periodicity & Poincare Sections

2-frequency dynamics: trajectories on surface of torus.

Control freq = R

:R r

r R

T pp q

T q

• q points

• skips over p-1 pts (clockwise)

Incommensurate

/ long period:

Drift ring

Ex 6.2-1

Quasi-Periodicity Route to Chaos

As parameter increases:• One f.p. (if not driven ).• Hopf bifurcation: f.p. → l.c.• 2nd freq (torus T2) → q.p. if

incommensurate.• 3rd freq (T3)• Small perturbation destroys torus → Chaos

C.f., Landau scheme

Universality in QP Route to ChaosReminder: 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

Observed

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.

Winding Numbers

Frequency-ratio parameter:

2

1

at vanishing non-linearity & coupling

Winding number:

2

1

W

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 1 W modulo 1W modulo 1 = fraction of circle trajectory travelled in 1 period of driving force.

Circle Map: Protype of QP

1n nf , 0, 1

Example:

1 1n n mod

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

.6

.2

2

1

mod 1

mod 1r

R

Poincare section:

Δn = 1 → 1 period of driving force

f m f m m Z

1 mod 1n nf

Winding number:

0 02

1

limn

n

fW

n

( no modulo 1 )

For the linear map

1 1n n mod

0 0nf n W

Ω rational → periodic

Ω irrational → quasi-periodic

0 0nf Increase ofΘin n units of

time =

Average frequency 2’ =

0 0limn

n

f

n

Sine-Circle Map

1 sin 2 12n n n

Kmod

0K

Ω = bare winding number

K = nonlinearity parameter

= frequency ratio parameter

Map is not invertible for K > 1→ folding → Chaos ?

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

,K0.95 ,0.5,K0.04 ,0.5,K0.5,2.0,K0.5,0.8Fixed points:

* * sin 2 * 12

Kmod

2sin 2 * m

K

0,1,m

2 m K

1 ' sin 2n n nK mod

1 cos2 0f K →1

cos2K

m = 0

m = 1 m = 2 K = 0.5

K = 2.0

Ω = 0.5 π π 3π no f.p.

no f.p.

Ω = 0.04 0.25 6.03 12.3 1 f.p.

Ω = 0.95 5.97 0.31 6.6 1 f.p.

Values of 2π|Ω-m|:

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

,K0.95 ,0.5,K0.04 ,0.5,K0.5,2.0,K0.5,0.80.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

1 cos2f K Stable if0 cos2 * 2K

Frequency-Locking

2sin 2 * m

K

→ * *f m

0 02

1

limn

n

fW

n

m

→ * *nf nm

* *lim

n

n

f

n

0,1,2,

→ m:1 frequency locking

0 cos2 * 2K 2 m K Stable fixed point for f:

→

Stable fixed point for f(k):

* *kf m → * *nkf nm

* *lim

nk

n

fW

nk

m

k 1 2

0, , ,k k

→ m:k frequency locking

K = 0.5, Ω = 0.04

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

1:2 locking K = 0.8, Ω = 0.5

f(2

)

0:1

1:1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

* *kf m → m:k frequency locking

K = 0.5, Ω = 0.94

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

Quasi-periodic K = 0.5, Ω = 0.6180339

Devil’s Staircase & Arnold’s Tongues

W vs Ω for K = 1 Devil’s staircase

Arnold’s tongues

, * *qKf p p:q

locking:

* sin 2 * *2

K

0:1 locking:

2 K

* sin 2 * * 12

K

1:1 locking:

2 1 K

0.55 0.6 0.65 0.7

0.5

0.55

0.6

0.65

0.7

0.75

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

Consider 2 frequency lockings p/q and p’/q’ at respective parameters Ω and Ω’ that are adjusted so that their fixed points coincide. qf p qf p

q q qf f f p

→

By definition, all circle maps are “periodic modulo”. f n f n for any Θ and

integer n

q qf f

(modulus action suspended)

i.e.

p p qp f

∴ 2 2f n f f n f n

q qf n f n

q qf p p Adjust Ω’’ to

get

f is monotonic with Ω → Ω’’ [Ω , Ω’]

Analytic Approach

Continued Fraction

0

1

23

111

G aa

aa

01 2 3

1 1 1a

a a a

0 1 2 3, , ,a a a a

nth order approximation:

0

1

23

111

n

n

G aa

aa a

0 0G a

01 2 3

1 1 1 1

n

aa a a a

0 1 2 3, , , , na a a a a

1 01

1G a

a 2 0

12

11

G aa

a

3 0

1

23

111

G aa

aa

Approximation of irrational number by rational fractions

0 0a 1ia for i = 1,2,3,…

11

11

11

G

1 1 1

1 1 1

1,1,1,

Golden Ratio

0 0G 1

11

1G 2

1 11 211

G

3

1 1 21 1 31 11 211

G

1

1

1nn

GG

→4

1 32 513

G

5

1 53 815

G

2 1 0G G

11 5

2G

11 5 0.618034...

2G 1 1

1 5 1.618034...2

G

G

G

1G

1

1

G G

G

1

1

1nn

GG

1

nn

n

FG

F

1

1

1 n

n

FF

1

n

n

F

F

1

n

n n

F

F F

1 1n n nF F F 0 0F 1 1F

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

0 0G 1

1

1G 2

1

2G 3

1 1 2

1 2 3G

4

1 2 3

2 3 5G

5

2 3 5

3 5 8G

6

3 5 8

5 8 13G

2 1 0G G

A continued fraction a0+(a1,a2,…) is periodic with period k if there exists a M such that

m m ka a for all m > M

Periodic fraction

solution to quadratic equation of integer coefficients

Ex.6.9-2,3: Silver mean (ai = 2)

= √2 - 1

Chaos & Universality

Sine circle map ( global features ):

• K → 0 : quasi-periodic ( W irrational ) for most Ω.

• K → 0 : fq.p. = c ( 1-K )β β ~ 0.34 (universal)

• K → 1- : frequency-locked ( W rational ) for most Ω.

• K = 1 : fq.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

Quasi-periodicity ( for K 1 ) : sequence of frequency lockings.e.g. Golden mean W via Fibonacci sequence { Wn = Fn/Fn+1 }. f’(0) = 0 for K

= 1Θ = 0 is part of the supercycles

1

nn

n

FW

F

Define Ωp/q (K) as the freq ratio that gives rise to W = p/q

Simplified notation for the case of the golden mean:

0 0limn

n

fW

n

is useless for calculations if W is

irrational

Sine circle map (local features)

Ωn gives 1 0n

n

FnKf F

i.e.

1

1

lim n n

nn n

K KK

K K

2

2.83362 1

11 5 2.6180339

12

K

for

K

1

lim n

nn

d KK

d K

dn(K) = distance from f(0) to nearest neighbor in the Ωn “supercycle”.

10n

n

Fn nKd K f F

1 0n

n

FnKf F

1 10n

n

FnKf F

nn K K C

aK N

Other forms of the scaling relations:

bK N

N a b

Golden mean

self 2.16443… 0.52687…

Silver mean

self 2.1748… 0.5239…

2 1

1

lim n n

nn n

W WN

W W

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

Universality Classes of Circle maps

Criteria: functional form near inflection point (Θ = 0 , K = 1).

12

zf K Prototyp

e:

1 1,

2 2

z ~ degree of inflection

z = 3 (cubic inflection) → δ,α same as sine circle mapLarge z: δ→ -4.11, α→ -1.0Ref: B.Hu, A.Valinai, O.Piro, Phys.Lett., A 144, 7-10 (90)

Summary

• Smears in B.D. → Q.P. (K < 1), chaos (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.

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 )

Forced Rayleigh-Benard ConvectionRef: J.Stavans, et al, PRL 55, 596 (85)

• R ~ convective flow

• r ~ B = 200 G

• K ~ I0

Mercury

I0 < 10 mA (circle-map-like):

freq-lock, q.p., Arnold tongues …

Golden mean seq → δ~ 2.8Silver mean seq → δ~ 7.0

Fractal dim of q.p. regions: see chap 9.

Breaking up of the torus in a Benard experiment

Ruelle-Takens route

Bernard Experiment

Periodically Perturbed Cardiac CellsGlass & 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) .

• Θn = g(Θn-1) + Ω (~ Sine circle map: f.l., q.p., A.T.,…)

q.p.

chaotic

Comments on Biological Models

• Models: rough guides.• Philosophical musings: pointless.• Too much details → missing the point.

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: δn(K) oscillates between -3.3 & -2.7 as n ↑.

– Explanation: actual freq-lock & chaos are different transitions better described by “integrate & fire” model.

2 0Q R Q Q Q

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

Forced van der Pol Oscillator

2 cosQ R Q Q Q F

V V I

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