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 - PowerPoint PPT Presentation
1 Universal Bicritical Behavior in Unidirectionally Coupled Systems Sang-Yoon Kim Department of Physics Kangwon National University Korea l Dynamical Systems ced Nonlinear Oscillators: Universal Period-Doubling Route to Chaos ly Coupled Systems ally coupled 1D maps, Unidirectionally coupled oscillators: a Model for Open Flow. d actively in connection with Secure Communication using Chaos Synch e universal scaling results for the 1D maps to the unidirectionally
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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
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
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Period-Doubling Transition to Chaos in The 1D Map
1D Map with A Single Quadratic Maximum
21 1)( ttt Axxfx
An infinite sequence of period doubling bifurcations ends at a finite accumulation point 506092189155401.1A
When exceeds , a chaotic attractor with positive appears.
A
tedtd )0()(
A
3
Critical Scaling Behavior near A=A
Parameter Scaling:
Orbital Scaling:
2669.4 ; largefor ~ nAA n
n
9502.2 ; largefor ~ nx n
n
Self-similarity in The Bifurcation Diagram
A Sequence of Close-ups(Horizontal and Vertical Magnification Factors: and )
1st Close-up 2nd Close-up
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Period-Doublings in Unidirectionally Coupled 1D Maps Unidirectionally Coupled 1D Maps
.1),(,1)(: 221
21 tttttttt CxByyxgyAxxfxT
Two Stability Multipliers of an orbit with period q determining the stability of the first and second subsystems:
.)2(,)2(1
21
1
q
tt
q
tt ByAx
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 ().
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Scaling Behavior near The Bicritical Point
Bicritical Point where two Feigenbaum critical lines meet Corresponding to a border of chaos in both subsystems
...)094090.1...,155401.1(),( cc BA
Scaling Behavior near (Ac, Bc)
1st subsystem
Feigenbaum critical behavior:
nn
ncn xAA 11 ~,~ ...)]502.2(...),669.4([ 11
...)601.1(**1,1 n
2nd subsystem
Non-Feigenbaum critical behavior:
nn
ncn yBB 22 ~,~ ]5053.1,3928.2[ 22
...)178.1(*2,2 n
~~
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Hyperchaotic Attractors near The Bicritical Point
)1.0(
,
BA
BBBAAA cc
2
1
BBB
AAA
c
c
22
21
BBB
AAA
c
c
02.0
121.0
2
1
04.0
242.0
2
1
01.0
061.0
2
1
~~
~~
~~
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Renormalization-Group (RG) Analysis of The Bicritical Behavior
Eigenvalue-Matching RG method
.1,1: 221
21),( tttttBA CxByyAxxT
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.”
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
),(),(),()( 1,2,21,1,1 BABAAA nnnn
Recurrence Relation between the Control Parameters A and B
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
5.0,2.0
(Ac, Bc)=(0.798 049 182 451 9, 0.802 377 2)
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Self-similar Topography of The Parameter Plane
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Hyperchaotic Attractors near The Bicritical Point
)111100.0,545100.0
,7003.0,85000.0(
,
*2
*1
xx
BA
BBBAAA cc
2
1
BBB
AAA
c
c
22
21
BBB
AAA
c
c
045.0
107.0
2
1
~~
023.0
055.0
2
1
~~
012.0
027.0
2
1
~~
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Bicritical Behavior in Unidirectionally Coupled Duffing Oscillators
);(),,(),(
),,,(,
122221222
11111
yyCtyxfyxxCyx
tyxfyyx
B
A
Eq. of Motion
tAxxytyxf A cos),,( 3
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
8.2,2.0
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Bicritical Behaviors in Unidirectionally Coupled Rössler Oscillators
)()(),(),(
),(,,
1222221222212222
1111111111
zzcxzbzyyayxyxxzyx
cxzbzayxyzyx
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
2.0ba
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Summary
Universal Bicritical Behaviors in A Large Class of Unidirectionally Coupled Systems
),( cc BAT
1
*T
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).
