1

Quasiperiodic Dynamics in Coupled Period-Doubling Systems

Sang-Yoon Kim

Department of Physics

Kangwon National University

Korea

Nonlinear Systems with Two Competing Frequencies

Mode Lockings, Quasiperiodicity, and Chaos

2

Symmetrically Coupled Period-Doubling Systems

Building Blocks: Period-Doubling Systems such as the 1D Map, Hénon Map, Forced Nonlinear Oscillators, and Autonomous Oscillators

Coupled Systems: Generic Occurrence of Hopf Bifurcations Quasiperiodic Transition

Purpose • Investigation of Mode Lockings, Quasiperiodicity, and Torus Doublings Associated with Hopf Bifurcations

• Comparison of the Quasiperiodic Behaviors of The Coupled Period-Doubling Systems with Those of the Circle Map (representative model for quasiperiodic systems with two competing frequencies)

3

~

~

R L

Ve= V0 sint

Rc

R L

R L

Ve= V0 sint

II

V0

L=470mH, f=3.87kHz, R=244 Single p-n junction resonator Period-doubling transition

Resistively coupled p-n junction resonators Quasiperiodic transition

L=100mH, Rc=1200, f=12.127kHz

V0

Quasiperiodic Transition in Coupled p-n Junction Resonators[ R.V. Buskirk and C. Jeffries, Phys. Rev. A 31, 3332 (1985). ]

Hopf Bifurcation

4

Quasiperiodic Transition in Coupled Parametrically Forced Pendulums (PFPs)

Single PFP

Symmetrically Coupled PFPs

Period-Doubling Transition

5.0,2.0

Quasiperiodic Transition

0.1C

.2sin)2cos(22),,(

,)(2

),,(,

,)(2

),,(,

2

2122222

1211111

xtAxtxxf

xxC

tyxfyyx

xxC

tyxfyyx

A

A

A

Hopf Bifurcation

5

Quasiperiodic Transition in Coupled Rössler Oscillators

Single Rössler Oscillator

Symmetrically Coupled Rössler Oscillators

Period-Doubling Transition

Quasiperiodic Transition

2.0ba

05.0

).()(

,,

),()(

,,

12222

222222

21111

111111

xxcxzbz

ayxyzyx

xxcxzbz

ayxyzyx

Hopf Bifurcation

6

Hopf Bifurcations in Coupled 1D Maps

.),(1

,),(1:

21

21

tttt

tttt

xygAyy

yxgAxxT

Two Symmetrically Coupled 1D Maps

Phase Diagram for The Linear Coupling Case with g(x, y) = C(y x)

Synchronous Periodic Orbits

Antiphase Orbits with Phase Shift ofHalf A Period (in a gray region)

Quasiperiodic Transition through A Hopf Bifurcation

Transverse PDB

7

Type of Orbits in Symmetrically Coupled 1D Maps

Exchange Symmetry:

Consider an orbit {zt}:

Strongly-Symmetric Orbits ()

Synchronous orbit on the diagonal ( = 0°)

Weakly-Symmetric Orbits (with even period n)

Antiphase orbit with phase shift of half a period () ( = 180°)

Asymmetric Orbits (, ) A pair of conjugate orbits {zt} and Dual Phase Orbits

),();(1 yxzzTz tt

),(),(; xyyxSTSTS

tt zzS

22/

nttn

t zzTzS

}{ tzS

1.0,5.1 CA

Symmetrically Coupled 1D Maps

Symmetry line: y = x (Synchronization line)

(In-phase Orbits)

8

Self-Similar Topography of The Antiphase Periodic Regimes

Antiphase Periodic Orbits in The Gray Regions

Self-Similarity near The Zero- Coupling Critical Point

Nonlinearity and coupling parameter scaling factors: (= 4.669 2…), (= 2.502 9…)

9

Hopf Bifurcations of Antiphase Orbits

Loss of Stability of An Orbit with Even Period n through A Hopf Bifurcation when its Stability Multipliers Pass through The Unit Circle at = e2i.

}{ *tz

Birth of Orbits with Rotation No. ( : Average Rotation Rate around a mother orbit point per period n of the mother antiphase orbit)

Quasiperiodicity (Birth of Torus)

irrational numbers Invariant Torus

Mode Lockings (Birth of A Periodic Attractor)

(rational no.) r / s (coprimes) Occurrence of Anomalous Hopf Bifurcations

r: even Symmetry-Conserving Hopf Bifurcation Appearance of a pair of symmetric stable and unstable orbits of rotation no. r / s r: odd Symmetry-Breaking Hopf Bifurcation Appearance of two conjugate pairs of asymmetric stable and unstable orbits of rotation no. r / s

18.0

,24.1

C

A

10

Arnold Tongues of Rotation No. (= r / s)

Unstable manifolds of saddle points flow into sinks, and thus union of sinks, saddles, and unstable manifolds forms a rational invariant circle.

A=1.266 and C= 0.196

A Pair of SymmetricSink and Saddle

Two Pairs of AsymmetricSinks and Saddles

52 83A=1.24 and C= 0.199

11

Bifurcations inside Arnold’s Tongues1. Period-Doubling Bifurcations

(Similar to the case of the circle map) Case of A Symmetric Orbit

Case of An Asymmetric Orbit

Hopf Bifurcation from The Antiphase Period-2 Orbit

(e.g. see the tongue of rot. no. 28/59)

(e.g. see the tongue of rot. no. 19/40)

12

2. Hopf Bifurcations Tongues inside Tongues

2nd Generation(daughter tongues inside their mother tongue of rot. no. 2/5)

3rd Generation(daughter tongues inside their mother tongue of rot. no. 4/5)

2/5

6/7

4/55/6

4/5

6/7

4/5

8/9

5/6

13

Transition from Torus to Chaos Accompanied by Mode Lockings

(Gradual Fractalization of Torus Loss of Smoothness)

Smooth Torus Wrinkled Torus Fractal Torus (Strange Nonchaotic Attractor) ? Mode Lockings Chaotic Attractor

(Wrinkling behavior of torus is masked by mode lockings.)

052.0,0

1.0,349.1

21

CA

044.0,0

1.0,3525.1

21

CA

012.0

,008.0

23/6

1.0,3526.1

2

1

CA

025.0,006.0

1.0,3532.1

21

CA

~ ~ ~

~

~

~

~~

009.0998.02 D 002.0895.02 D

004.0123.12 D

14

Quasiperiodic Dynamics in Coupled 1D Maps

Hopf Bifurcations of Antiphase Orbits

Quasiperiodicity (invariant torus) + Mode Lockings

Question: Coupled 1D Maps may become a representative model for the quasiperiodic behavior in symmetrically coupled system?

No !

15

Torus Doublings in Symmetrically Coupled Oscillators Occurrence of Torus Doublings in Coupled Parametrically Forced Pendulums ( = 0.2, = 0.5, and A = 0.352)

48.0C 517.0C

524.0C 534.0C

DoubledTorus

16

Torus Doublings in Coupled Hénon Maps

,,,1

,,,1:

1

2

1

1

2

1

ttttttt

ttttttt

buvxugAuvu

bxyuxgAxyxT

.,

10

xuCuxg

b

Symmetrically Coupled Hénon Maps

Torus doublings may occur only in the (invertible) N-D maps (N 3).

Characterization of Torus Doublings by The Spectrum of Lyapunov Exponents

17

Torus Doublings for b = 0.5 and A = 2.05

29.0C 305.0C

31.0C 34.0C

reverse normal

18

Damping Effect on Torus Doublings and Mode Lockings

Torus doublings occur for b > 0.3. (No torus doublings for b < 0.3)

As b is increased, the region of mode lockings decreases.

~

b = 0.7b = 0.5b = 0.3

19

b 0.3 0.5 0.7

Hyperchaos 24.36 % 20.64 % 12.03 %

Chaos 37.55 % 34.86 % 19.88 %

Mode lockings 15.66 % 13.52 % 11.13 %

Torus 22.40 % 27.81 % 48.19 %

Doubled torus 0.03 % 3.17 % 8.17 %

Period-4 torus 0 % 0 % 0.60 %

Ratios of Hyperchaos, Chaos, Torus, and Mode Lockings

20

Summary

In Symmetrically Coupled Period-Doubling Systems, Mode Lockings and

Quasiperiodicity occur through Hopf Bifurcations of Antiphase Orbits

(Representative model: Coupled 1D Maps).

Bifurcations inside the Arnold tongues become richer than those in the case of the

circle map

Torus Doublings also occur in Symmetrically Coupled Hénon Maps when the

damping parameter becomes larger than a threshold value, which is in contrast

to the coupled 1D maps without torus doublings.

Effect of Asymmetry on The Quasiperiodic Behavior

10,),(1

,),()1(1:

21

21

tttt

tttt

xygAyy

yxgAxxT

Threshold value *, s.t. 0 < < * Robustness of The Quasiperiodic Behavior

> * No Hopf Bifurcation (No Quasiperiodic Behavior)

21

Complex Dynamics in Symmetrically Coupled Systems

In-phase orbits Universal Scalings of Period Doublings

Antiphase orbits Quasiperiodic Dynamics (Hopf Bif.)

Dual phase orbits What’s their dynamics?

Period Doublings

Feigenbaum lines

Scaling near both end?

22

Stability Analysis in Coupled Hénon Maps

,,),(1

,,),(1:

12

1

12

1

ttttttt

ttttttt

buvxugAuvu

bxyuxgAxyxT

qbDMMBMADDABAP 222

21234 ,]Tr Tr [ ,Tr ;

Complex Quadruplet: *,,, *

DD Hopf Bifurcation

.)(),( xuCuxg

Consider an orbit of period q. Its stability is determined by its Stability Multipliers which are the eigenvalues of the linearized map M (=DTq) of Tq around the period-q orbit.

M: Dissipative Symplectic Map Eigenvalues come into pairs lying on the circle of radius D1/4