1. 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. 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. 3 ~ ~ R L V e = V 0 sin  t RcRc RL R L I I V0V0 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, R c =1200 , f=12.127kHz V0V0 Quasiperiodic Transition in Coupled p-n Junction Resonators [ R.V. Buskirk and C. Jeffries, Phys. Rev. A 31, 3332 (1985). ] Hopf Bifurcation

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

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

6. 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. 7 Type of Orbits in Symmetrically Coupled 1D Maps Exchange Symmetry: Consider an orbit {z t }:  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 {z t } and Dual Phase Orbits  Symmetrically Coupled 1D Maps  Symmetry line: y = x (Synchronization line) (In-phase Orbits)

8. 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. 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. 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 Symmetric Sink and Saddle Two Pairs of Asymmetric Sinks and Saddles A=1.24 and C=  0.199

11. 11 Bifurcations inside Arnold’s Tongues 1. 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. 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 4/5 5/6 4/5 6/7 4/5 8/9 5/6

13. 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. 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. 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. 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. 17  Torus Doublings for b = 0.5 and A = 2.05 reverse normal

18. 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 = 0.7 b = 0.5 b = 0.3

19. 19 b0.30.50.7 Hyperchaos24.36 %20.64 %12.03 % Chaos37.55 %34.86 %19.88 % Mode lockings15.66 %13.52 %11.13 % Torus22.40 %27.81 %48.19 % Doubled torus 0.03 % 3.17 % 8.17 % Period-4 torus 0 % 0.60 %  Ratios of Hyperchaos, Chaos, Torus, and Mode Lockings

20. 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. 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. 22 Stability Analysis in Coupled Hénon Maps Complex Quadruplet:  Hopf Bifurcation Consider an orbit of period q. Its stability is determined by its Stability Multipliers which are the eigenvalues of the linearized map M (=DT q ) of T q around the period-q orbit. M: Dissipative Symplectic Map  Eigenvalues come into pairs lying on the circle of radius D 1/4