Complex Dynamics in Coupled Systems: Quasiperiodicity, Torus Doublings, and Mode Locking

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 Chaos Coupled Systems: Generic Occurrence of Hopf Bifurcations  Mode Locking, Quasiperiodicity, Torus Doubling, Chaos, Hyperchaos

2. 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, R=244, f=3.87kHz R L I Ve= V0 sin(2ft) ~ 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(2ft) ~ V0

3. Quasiperiodic Transition in Coupled Pendula  Single Pendulum Period-Doubling Transition  Symmetrically Coupled Pendula Hopf Bifurcation Quasiperiodic Transition

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

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

6. Type of Orbits in Symmetrically Coupled 1D Maps  Symmetrically Coupled 1D Maps Exchange Symmetry: STS = T; S(x,y) = (y,x)  Symmetry line: y = x (Synchronization line) Consider an orbit {zt}: • Strongly-Symmetric Orbits () Szt = zt (In-phase Orbits) Synchronous orbit on the diagonal ( = 0°)  Weakly-Symmetric Orbits (with even period n) • Szt = Tn/2zt = zt+n/2 • Antiphase orbit with phase shift of • half a period () ( = 180°) •  Asymmetric Orbits (, ) • A pair of conjugate orbits {zt} and {Szt} • Dual Phase Orbits [Periodic Orbits can be Classified in Terms of the Periods and Phase Shifts (p, q): q = 0, …, p-1]

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  = 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  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  

9. Arnold Tongues of Rotation No.  (= r/s)  Standard Hopf Bifurcation A Pair of Symmetric Sink and Saddle a=1.266 and = -0.169  Nonstandard Double Hopf Bifurcation Two Conjugate Pairs of Asymmetric Sinks and Saddles a=1.24 and = -0.2

10. Bifurcation Patterns 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-4 Orbit PDB SNB PFB SNB (e.g. see the tongue of rot. no. 18/47)  Case of an Asymmetric Orbit SNB PDB SNB (e.g. see the tongue of rot. no. 17/44)

11. Tongues inside Tongues 2. Hopf Bifurcations 2nd Generation (daughter tongues inside their mother tongue of rot. no. 2/5) HB SNB (e.g. see the tongue of rot. no. 2/5) 3. Period-Doubling and Hopf Bifurcations SNB PDB HB 3rd Generation (daughter tongues inside their mother tongue of rot. no. 4/5) SNB PFB (e.g. see the tongue of rot. no. 12/31)

12. 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.)

13. Effect of Asymmetry on Hopf Bifurcations  System =0: symmetric coupling =1: unidirectional coupling 0<* (=0.392) : Hopf Bifurcations Leading to Quasiperiodicity and Mode Locking *<1: Period-Doubling Bifurcations

14. Hopf Bifurcations in Coupled Pendula  Nonstandard Double Hopf Bifurcation  Standard Hopf Bifurcation A=2.75 and = -1.156 A=2.75 and = -1.11

15. 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)

16. Torus Doublings in Symmetrically Coupled Pendula • Occurrence of Torus Doublings in Symmetrically Coupled Pendula ( = 0.1,  = 1, and = 0.6) Doubled Torus

17. Torus Doublings in Coupled Hénon Maps • 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 

18.  Torus Doublings for b = 0.5 and  = -0.305

19. Damping Effect on the Ratios of Dynamical Regimes b = 0.7 b = 0.5 b = 0.2 Doubled Torus Doubled Torus  Occurrence of Torus Doublings for b > 0.3 ~  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

20. 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 Universality Confirmed in Symmetrically Coupled Period-Doubling Oscillators such as the Coupled Pendula and Rössler Oscillators

21. 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 zt: Fixed Point of Both Tn and R Tn=RR (R: Half-Cycle Map)  Standard Hopf Bifurcation in R Stability Multiplier:  = e 2ip/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 2ir/s = e 2i (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)