1

Complex Dynamics in Coupled Period-Doubling Systems;Mode Locking, Quasiperiodicity, and Torus Doublings

Sang-Yoon Kim

Department of Physics

Kangwon National University

Nonlinear Systems with Two Competing Frequencies (e.g. circle map)

Mode Lockings, Quasiperiodicity, and Chaos

Coupled Period-Doubling Systems (e.g. coupled logistic maps)

Single System: Period-Doubling Transition to ChaosCoupled Systems: Generic Occurrence of Hopf Bifurcations Mode Locking, Quasiperiodicity, Torus Doubling, Chaos, Hyperchaos

2

~

~

R L

Ve= V0 sin(2ft)

Rc

R L

R L

Ve= V0 sin(2ft)

II

V0

L=470mH, R=244, f=3.87kHz

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

3

Quasiperiodic Transition in Coupled Pendula

Single Pendulum

Symmetrically Coupled Pendula

Period-Doubling Transition

1,7.0

Quasiperiodic Transition

8.0

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

,)(),,(,

,)(),,(,

2

2122222

1211111

xtAxtxxf

xxtyxfyyx

xxtyxfyyx

A

A

A

Hopf Bifurcation

4

Quasiperiodic Transition in Coupled Rössler Oscillators

Single Rössler Oscillator

Symmetrically Coupled Rössler Oscillators

Period-Doubling Transition

Quasiperiodic Transition

2.0ba

1.0

).(,

),(

),(,

),(

222222

12222

111111

21111

cxzbzayxy

xxzyx

cxzbzayxy

xxzyx

Hopf Bifurcation

5

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) = (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

a

6

Type of Orbits in Symmetrically Coupled 1D Maps

Exchange Symmetry: STS = T; S(x,y) = (y,x)

Consider an orbit {zt}:

Strongly-Symmetric Orbits () Szt = zt (In-phase Orbits)

Synchronous orbit on the diagonal ( = 0°)

Weakly-Symmetric Orbits (with even period n) Szt = Tn/2 zt = zt+n/2

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

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

),();(1 yxzzTz tt

01.0,31.1 a

Symmetrically Coupled 1D Maps

Symmetry line: y = x (Synchronization line)

[Periodic Orbits can be Classified in Terms of the Periods and Phase Shifts (p, q): q = 0, …, p1]

7

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…)

8

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 Standard Hopf Bifurcation Appearance of a pair of symmetric stable and unstable orbits of rotation no. r / s r: odd Nonstandard Double Hopf Bifurcation Appearance of two conjugate pairs of asymmetric stable and unstable orbits of rotation no. r / s

25.0

,2.1

a

9

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

a=1.266 and = 0.169

A Pair of SymmetricSink and Saddle

Two Conjugate Pairs of Asymmetric

Sinks and Saddles

52

83

a=1.24 and = 0.2

Standard Hopf Bifurcation

Nonstandard Double Hopf Bifurcation

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Bifurcation Patterns 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-4 Orbit

(e.g. see the tongue of rot. no. 18/47)

(e.g. see the tongue of rot. no. 17/44)

SNB

PFB

PDB

SNB

PDB

SNB

SNB

11

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)

3. Period-Doubling and Hopf Bifurcations

HB

SNB

SNB

HB SNB

PFB

PDB

(e.g. see the tongue of rot. no. 2/5)

(e.g. see the tongue of rot. no. 12/31)

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Transition from Torus to Chaos Accompanied by Mode Lockings

Smooth Torus Wrinkled Torus Mode Lockings Chaotic Attractor

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

23.1a

24.1a

238.1a

26.1a

008.0

8/3

1

2.0

01 01

038.01

13

Effect of Asymmetry on Hopf Bifurcations

.)(1

,)()1(1:

21

21

tttt

tttt

yxcayy

xycaxxT

System

=0: symmetric coupling=1: unidirectional coupling

0<* (=0.392) : Hopf Bifurcations Leading to Quasiperiodicity and Mode Locking*<1: Period-Doubling Bifurcations

0 1

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Hopf Bifurcations in Coupled Pendula

Standard Hopf Bifurcation Nonstandard Double Hopf Bifurcation

1,7.0

A=2.75 and = 1.11

9/2

A=2.75 and = 1.156

5/1

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Dynamical Behaviors of Symmetrically Coupled 1D Maps

Hopf Bifurcations of Antiphase Orbits

Quasiperiodicity (invariant torus) + Mode Lockings

Another Interesting Behavior of Symmetrically Coupled Oscillators: Torus Doublings (no occurrence in coupled 1D maps)

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Torus Doublings in Symmetrically Coupled Pendula Occurrence of Torus Doublings in Symmetrically Coupled Pendula ( = 0.1, = 1, and = 0.6)

DoubledTorus

016.0,008.0,5174.1 21 A

037.0,008.0,517.1 21 A

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Torus Doublings in Coupled Hénon Maps

),()1(,)(),()(1)()1(

),()1(,)(),()(1)()1(:

22122222

11212111

tbxtytxtxgtaxtytx

tbxtytxtxgtaxtytxT .,

10

1221 xxxxg

b

Symmetrically Coupled Hénon Maps (Representative Model for Poincaré Maps of Coupled Period-Doubling Oscillators)

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

Characterization of Torus Doublings by the Spectrum of Lyapunov Exponents

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Torus Doublings for b = 0.5 and = -0.305

024.0,0,04.2 21 a 017.0,0,05.2 21 a

01.0,007.0,07.2 21 a

27/4

031.0,049.0,08.2 21 a

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Damping Effect on the Ratios of Dynamical Regimes

Occurrence of Torus Doublings for b > 0.3~

b = 0.7b = 0.5b = 0.2

Doubled Torus

DoubledTorus

Increase in the Ratios of the Smooth Torus(T), Doubled Torus(2T), and Quadrupled Torus(4T)

Decrease in the Ratios of the Mode Locking, Chaos, and Hyperchaos

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Summary

Mode Lockings and Quasiperiodicity via Hopf Bifurcations of Antiphase Orbits in Coupled 1D Maps

Occurrence of Anomalous Hopf Bifurcations; Standard and Nonstandard Double Hopf Bifurcations

Occurrence of Torus Doublings in Symmetrically Coupled Hénon Maps for b > b*

(in contrast to the coupled 1D maps without torus doublings)

Effect of the Asymmetry of Coupling on Hopf Bifurcations

nsBifurcatio Doubling-Period :1

s)(RobustnesnsBifurcatio Hopf :0 s.t. valueThreshold*

**

Universality

Confirmed in Symmetrically Coupled Period-Doubling Oscillators such as the Coupled Pendula and Rössler Oscillators

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Anomalous Hopf Bifurcations of Antiphase Orbits

Anitphase Orbits

{zt = (xt, yt)}: Antiphase Orbit with an Even Period n in Coupled 1D Maps T with the Exchange Symmetry S

ttn

tnttn

tttn zzTSzRzzTzSzzT )()()()(,)( 2/

2/2/

zt: Fixed Point of Both Tn and R

Tn=RR (R: Half-Cycle Map)

Standard Hopf Bifurcation in RStability Multiplier: = e 2i p/q Appearance of a Pair of Stable and Unstable Orbits of Rotation Number R (=p/q)

Anomalous Hopf Bifurcation in Tn

Stability Multiplier of zt in Tn: = e 2i r/s = e 2i (2p)/q

(1) Standard Hopf Bifurcation (q: odd r: even) r = 2p (even), s = q (odd) Appearance of a Pair of Stable and Unstable Orbits of Rotation Number (=r/s)

(2) Nonstandard Double Hopf Bifurcation (q: even r: odd) r = p (odd), s = q/2 Appearance of Two Pairs of Stable and Unstable Orbits of Rotation Number (=r/s)