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

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

4

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 ().

5

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

~~

6

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

~~

~~

~~

7

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

’’’

’’

Self-similar(A, B) ’ ’(A , B )

8

Fixed Point and Relevant Eigenvalues

Fixed Point (A*, B*) Bicritical Point (Ac, Bc)

),(),(),()( **1,2

**2

*1,1

*,1 BABAAA nnn

*

,2

*

,1 ,dy

dy

dx

dxnn

Orbital Scaling Factors

*

,2

*

1,2,2

*

,1

*

1,1,1 ,

dB

d

dB

d

dB

dB

dA

d

dA

d

dA

dA nnn

nnn

Relevant Eigenvalues

’’

’ ’ ’ ’

9

RG Results

n

11 1.401 155 189 092 050 6 1.090 094 348 817

12 1.401 155 189 092 050 6 1.090 094 348 536

13 1.401 155 189 092 050 6 1.090 094 348 675

14 1.401 155 189 092 050 6 1.090 094 348 704

15 1.401 155 189 092 050 6 1.090 094 348 701

1.401 155 189 092 050 6 1.090 094 348 701

Bicritical point

n 1,n 2,n

11 4.669 201 609 1 2.392 81

12 4.669 201 609 1 2.392 78

13 4.669 201 609 1 2.392 74

14 4.669 201 609 1 2.392 73

15 4.669 201 609 1 2.392 73

4.669 201 609 1 2.392 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

*nA *

nB

Orbital scaling factors

10

Unidirectionally Coupled Parametrically Forced Pendulums

Parametrically Forced Pendulum (PFP)

Normalized Eq. of Motion:

xtAx

txxfx

2sin)2cos(22

),,(2

Unidirectionally Coupled PFPs

).(),,(),(

),,,(,

122221222

11111

yyCtyxfyxxCyx

tyxfyyx

B

A

)cos()( tth

O

S

l

m

),,( tyxxfy

yx

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

5.0,2.0

(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

)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

~~

14

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

15

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

16

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).

)6692.4(1 : Feigenbaum constant

3928.22 : Non-Feigenbaum constant~

2

),( BAT

(scaling factor in the drive subsystem)

(scaling factor in the response subsystem)