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

3. 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:  and  ) 1st Close-up 2nd Close-up

4. 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 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. 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 (A c, B c ) 1st subsystem Feigenbaum critical behavior: 2nd subsystem Non-Feigenbaum critical behavior:  ~  ~  

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

7. 7 Renormalization-Group (RG) Analysis of The Bicritical Behavior  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=2 n, 2 n+1 ) become “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 ’ ’ ’   ’ ’ Self-similar (A, B) ’ ’

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

9. 9 RG Results n 111.401 155 189 092 050 61.090 094 348 817 121.401 155 189 092 050 61.090 094 348 536 131.401 155 189 092 050 61.090 094 348 675 141.401 155 189 092 050 61.090 094 348 704 151.401 155 189 092 050 61.090 094 348 701 1.401 155 189 092 050 61.090 094 348 701  Bicritical point n  1,n  2,n 114.669 201 609 12.392 81 124.669 201 609 12.392 78 134.669 201 609 12.392 74 144.669 201 609 12.392 73 154.669 201 609 12.392 73 4.669 201 609 12.392 7 n  1,n  2,n 11  2.502 907 744 9  1.505 163 12  2.502 907 847 2  1.505 263 13  2.502 907 869 1  1.505 280 14  2.502 907 873 8  1.505 296 15  2.502 907 874 8  1.505 311  2.502 907 875 1  1.505 318  Parameter scaling factors  Orbital scaling factors

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

11. 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 (A c, B c ) Same as that in the abstract system of unidirectionally-coupled 1D maps (A c, B c )=(0.798 049 182 451 9, 0.802 377 2)

12. 12 Self-similar Topography of The Parameter Plane

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

14. 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. 15 Bicritical Behaviors in Unidirectionally Coupled Rössler Oscillators  Eq. of Motion c 1 & c 2 : 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. 16 Summary  Universal Bicritical Behaviors in A Large Class of Unidirectionally Coupled Systems    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). : Feigenbaum constant : Non-Feigenbaum constant  ~ (scaling factor in the drive subsystem) (scaling factor in the response subsystem)