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Explore period-doubling route to chaos in unidirectionally coupled systems using 1D maps and nonlinear oscillators to understand scaling behavior. The study involves Feigenbaum critical lines, stability diagrams, and hyperchaotic attractors near the bicritical point.
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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
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.
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
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 ().
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:
Hyperchaotic Attractors near The Bicritical Point ~ ~ ~ ~ ~ ~ - - - - - -
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
Fixed Point and Relevant Eigenvalues Fixed Point (A*, B*) Bicritical Point (Ac, Bc) Relevant Eigenvalues ’ ’ ’ ’ Orbital Scaling Factors ’ ’
RG Results Bicritical point Parameter scaling factors Orbital scaling factors
Unidirectionally Coupled Parametrically Forced Pendulums Parametrically Forced Pendulum (PFP) O Normalized Eq. of Motion: S l m Unidirectionally Coupled PFPs
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)
Hyperchaotic Attractors near The Bicritical Point ~ ~ ~ ~ ~ ~ - - - - - -
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
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
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).