Universal Bicritical Behavior in Unidirectionally Coupled Systems

1. Universal Bicritical Behavior in Unidirectionally Coupled Systems • Sang-Yoon Kim • Department of Physics • Kangwon National University • Korea • Low-dimensional Dynamical Systems 1D Maps, Forced Nonlinear Oscillators: Universal Period-Doubling Route to Chaos • UnidirectionallyCoupled Systems • Unidirectionally coupled 1D maps, Unidirectionally coupled oscillators: • Used as a Model for Open Flow. • Discussed actively in connection with Secure Communication using Chaos Synchronization • Purpose • To extend the universal scaling results for the 1D maps to the unidirectionally coupled systems

2.  An infinite sequence of period doubling bifurcations ends at a finite accumulation point Period-Doubling Transition to Chaos in The 1D Map  1D Map with A Single Quadratic Maximum  When exceeds , a chaotic attractor with positive s appears.

3. Critical Scaling Behavior near A=A  Parameter Scaling:  Orbital Scaling:  Self-similarity in The Bifurcation Diagram A Sequence of Close-ups (Horizontal and Vertical Magnification Factors: d and a) 2nd Close-up 1st Close-up

4. Period-Doublings in Unidirectionally Coupled 1D Maps  Unidirectionally Coupled 1D Maps  Two Stability Multipliers of an orbit with period q determining the stability of the first and second subsystems: Period-doubling bif. Saddle-node bif.  1 -1  Stability Diagram of the Periodic Orbits Born via PDBs for C = 0.45. Vertical dashed line: Feigenbaum critical line for the 1st subsystem Non-vertical dashed line: Feigenbaum critical line for the 2nd subsystem Two Feigenbaum critical lines meet at the Bicritical Point ().

5. Scaling Behavior near The Bicritical Point ~ ~ - -  Bicritical Point where two Feigenbaum critical lines meet Corresponding to a border of chaos in both subsystems  Scaling Behavior near (Ac, Bc) 1st subsystem  Feigenbaum critical behavior: 2nd subsystem  Non-Feigenbaum critical behavior:

6. Hyperchaotic Attractors near The Bicritical Point ~ ~ ~ ~ ~ ~ - - - - - -   

7. Renormalization-Group (RG) Analysis of The Bicritical Behavior ’ ’ (A , B )  Eigenvalue-Matching RG method Basic Idea: ’ ’ For each parameter-value (A, B) of level n, associate a parameter-value (A , B ) of the next level n+1 such that periodic orbits of level n and n+1 (period q=2n, 2n+1) become “self-similar.”  (A, B) Self-similar Orbit of level n Orbit of level n+1 A simple way to implement the basic idea is to equate the SMs of level n and n+1  ’ ’ ’ Recurrence Relation between the Control Parameters A and B

8. Fixed Point and Relevant Eigenvalues  Fixed Point (A*, B*)  Bicritical Point (Ac, Bc)  Relevant Eigenvalues ’ ’ ’ ’  Orbital Scaling Factors ’ ’

9. RG Results  Bicritical point  Parameter scaling factors  Orbital scaling factors

10. Unidirectionally Coupled Parametrically Forced Pendulums  Parametrically Forced Pendulum (PFP) O Normalized Eq. of Motion: S  l m  Unidirectionally Coupled PFPs

11. Stability Diagram of Periodic Orbits for C = -0.2  Structure of the stability diagram Same as that in the abstract system of unidirectionally-coupled 1D maps  Bicritical behavior near (Ac, Bc) Same as that in the abstract system of unidirectionally-coupled 1D maps (Ac, Bc)=(0.798 049 182 451 9, 0.802 377 2)

12. Self-similar Topography of The Parameter Plane

13. Hyperchaotic Attractors near The Bicritical Point ~ ~ ~ ~ ~ ~ - - - - - -   

14. Bicritical Behavior in Unidirectionally Coupled Duffing Oscillators  Eq. of Motion A & B: Control parameters of the 1st and 2nd subsystems, C: coupling parameter  Stability Diagram for C = -0.1 Antimonotone Behavior Forward and Backward Period- Doubling Cascades Structure of the stability diagram Same as that in the abstract system of unidirectionally-coupled 1D maps Bicritical behaviors near the four bicritical points Same as those in the abstract system of unidirectionally-coupled 1D maps   

15. Bicritical Behaviors in Unidirectionally Coupled Rössler Oscillators  Eq. of Motion c1 & c2: Control parameters of the 1st and 2nd subsystems, : coupling parameter  Stability Diagram for = -0.01 Structure of the stability diagram Same as that in the abstract system of unidirectionally-coupled 1D maps Bicritical behavior near bicritical point Same as that in the abstract system of unidirectionally-coupled 1D maps  

16. Summary : Feigenbaum constant ~ - : Non-Feigenbaum constant  Universal Bicritical Behaviors in A Large Class of Unidirectionally Coupled Systems (scaling factor in the drive subsystem)   (scaling factor in the response subsystem)  Eigenvalue-matching RG method is a very effective tool to obtain the bicritical point and the scaling factors with high precision.  Bicritical Behaviors: Confirmed in Unidirectionally Coupled Oscillators consisting of parametrically forced pendulums, double-well Duffing oscillators, and Rössler oscillators Refs: 1. S.-Y. Kim, Phys. Rev. E 59, 6585 (1999). 2. S.-Y. Kim and W. Lim, Phys. Rev E 63, 036223 (2001). 3. W. Lim and S.-Y. Kim, AIP Proc. 501, 317 (2000). 4. S.-Y. Kim, W. Lim, and Y. Kim, Prog. Theor. Phys. 106, 17 (2001).