Quasiperiodic Dynamics in Coupled Period-Doubling Systems

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. Quasiperiodic Transition in Coupled p-n Junction Resonators [ R.V. Buskirk and C. Jeffries, Phys. Rev. A 31, 3332 (1985). ]  Single p-n junction resonator  Period-doubling transition L=470mH, f=3.87kHz, R=244 R L I Ve= V0 sint ~ V0  Resistively coupled p-n junction resonators  Quasiperiodic transition L=100mH, Rc=1200,f=12.127kHz R L Hopf Bifurcation I Rc R L Ve= V0 sint ~ V0

4. Quasiperiodic Transition in Coupled Parametrically Forced Pendulums (PFPs)  Single PFP Period-Doubling Transition  Symmetrically Coupled PFPs Hopf Bifurcation Quasiperiodic Transition

5. Quasiperiodic Transition in Coupled Rössler Oscillators  Single Rössler Oscillator Period-Doubling Transition  Symmetrically Coupled Rössler Oscillators Hopf Bifurcation Quasiperiodic Transition

6. Hopf Bifurcations in Coupled 1D Maps  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 of Half A Period (in a gray region) Quasiperiodic Transition through A Hopf Bifurcation Transverse PDB

7. Type of Orbits in Symmetrically Coupled 1D Maps  Symmetrically Coupled 1D Maps Exchange Symmetry:  Symmetry line: y = x (Synchronization line) 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 (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  = e2i. 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  

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 Pair of Symmetric Sink and Saddle Two Pairs of Asymmetric Sinks and Saddles A=1.24 and C= -0.199 A=1.266 and C= -0.196

11. Bifurcations inside Arnold’s Tongues 1. Period-Doubling Bifurcations (Similar to the case of the circle map)  Case of A Symmetric Orbit Hopf Bifurcation from The Antiphase Period-2 Orbit (e.g. see the tongue of rot. no. 28/59)  Case of An Asymmetric Orbit (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) 6/7 2/5 5/6 4/5 8/9 6/7 3rd Generation (daughter tongues inside their mother tongue of rot. no. 4/5) 5/6 4/5 4/5

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.)

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) Doubled Torus

16. Torus Doublings in Coupled Hénon Maps  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 reverse normal

18. Damping Effect on Torus Doublings and Mode Lockings b = 0.3 b = 0.5 b = 0.7  Torus doublings occur for b > 0.3. (No torus doublings for b < 0.3)  As b is increased, the region of mode lockings decreases. ~

19.  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 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 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  Hopf Bifurcation Complex Quadruplet: