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

9 RG Results n  Bicritical point n  1,n  2,n n  1,n  2,n 11              Parameter scaling factors  Orbital scaling factors

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

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 )=( , )

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 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 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, (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)