1

Universal Critical Behavior of Period Doublings in Symmetrically Coupled Systems

Sang-Yoon Kim

Department of Physics

Kangwon National University

Korea

Low-dimensional Dynamical Systems (1D Maps, Forced Nonlinear Oscillators)

Universal Routes to Chaos via Period Doublings, Intermittency, and Quasiperiodicity: Well Understood

Coupled High-dimensional Systems (Coupled 1D Maps, Coupled Oscillators)

Coupled Systems: used to model many physical, chemical, and biological systems such as Josephson junction arrays, chemical reaction-diffusion systems, and biological-oscillation systems

Purpose To investigate critical scaling behavior of period doubling in coupled systems and to extend the results of low-dimensional systems to coupled high-dimensional systems.

2

Period-doubling Route 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

Renormalization-Group (RG) Analysis of The Critical Behavior

RG operator

.)1(1;))(()()()(1 nnnnnnnn fxffxfxf

(fn: n-times renormalized map)

Squaring operator Looking at the system on the doubled time scale

Rescaling operator Making the new (renormalized) system as similar to the old system as possible

Attraction of the critical map to the fixed map with one relevant eigenvalue Af *f

;)(lim *ff An

n

)()()( ** xfxf

Af

*f’

5

Critical Behavior of Period Doublings in Two Coupled 1D Maps

.),()(),(

,),()(),(:

1

1

tttttt

tttttt

xygyfxyFy

yxgxfyxFxT

Two Symmetrically Coupled 1D Maps

,1)( 2Axxf g(x,y): coupling function satisfying a condition g(x,x) = 0 for any x

Exchange Symmetry Invariant Synchronization Line y = x

Synchronous (in-phase) orbits on the y = x line

Asynchronous (out-of-phase) orbits

Concern Critical scaling behavior of period doublings of synchronous orbits

6

Stability Analysis of Synchronous Periodic Orbits

Two stability multipliers for a synchronous orbit of period q:

Longitudinal Stability Multiplier | |

Determining stability against the longitudinal perturbation along the diagonal

1

0|| )(

q

ttxf : Same as the 1D stability multiplier

Transverse Stability Multiplier Determining stability against the transverse perturbation across the diagonal

;)(2)(1

0

q

ttt xGxf

xyy

yxgxG

),(

(Reduced coupling function)’

’

Period-doubling bif. Saddle-node bif.

1 1

Period-doubling bif. Pitchfork bif.

1 1

7

Renormalization-Group (RG) Analysis for Period Doublings

Period-doubling RG operator for the symmetrically 1D maps T

.0

0

B

:0

TT n

n

.),()(),(

,),()(),(:

1

1

ttntnttnt

ttntnttntn xygyfxyFy

yxgxfyxFxT

RG Eqs. for the uncoupled part f and the coupling part g:

))(()(1 xffxf nnn

.))(()),(,),((),(1 xffxyFyxFFyxg nnnnnn

n-times renormalized map,

(RG Eq. for the 1D case),

Reduced period-doubling RG operator

Def: Reduced Coupling Function xy

y

yxgxG

),(

)())(()())((2))(()(1 xfxfGxGxfGxffxG nnnnnnnnn

,)( 12 BTBT

),(),( 11 nnnn GfGf

’ ’

Note that keeps all the essential informations contained in .

[It’s not easy to directly solve the Eq. for the coupling fixed function g*(x, y).]

8

Fixed Points of and Their Relevant Eigenvalues Three fixed points (f *,G*) of (f *,G*) = (f *,G*): fixed-point Eq. f *(x): 1D fixed function, G*(x): Reduced coupling fixed function

G*(x) c (CE) (CTSM)

I 0

II 2 1

III Nonexistent 0

2,...)502.2( 21 )1)(( *

21 xf

)(*21 xf

...)601.1(*

*

Relevant eigenvalues of fixed pointsReduced Linearized Operator

n

n

n

n

n

n hhh

23

1

1

1 0

Note the reducibility of into a semi-block form)()(][ **

1 xhxh

)()()(),()()( ** xxGxGxhxfxf

)()(][ **2 xx c

Critical stability multipliers (SMs)

One relevant eigenvalue (=4.669…) (1D case): Common Eigenvalue c: Coupling eigenvalue (CE)

For the critical case, : SMs of an orbit of period 2n : Critical SMs

...)601.1()ˆ( ***|| xf

)ˆ(ˆ;)ˆ(2)ˆ( **** xfxxGxf

(1D critical SM): Common SM

),( ,||, nn ),( **|| n

’

’

’

’

9

Critical Scaling Behaviors of Period Doublings

1. Linearly-coupled case with g(x, y) = c(y x)

Stability Diagram for The Synchronous Orbits

Asymptotic Rule for The Tree Structure

1. U branching Occurrence only at the zero c-side (containing the zero-coupling point)

2. Growth like a “chimney” Growth of the other side without any further branchings

Bifurcation Routes 1. U-route converging to the zero-coupling critical point 2. C-routes converging to the critical line segments

Critical set Zero-coupling Critical Point + an Infinity of Critical Line Segments

10

A. Scaling Behavior near The Zero-Coupling Critical Point

Governed by the 1st fixed point GI = 0 with two relevant CE’s 1 = (-2.502 …) and 2=2.

CTSM: | | = = * (=1.601…)

Scaling of The Nonlinearity and Coupling Parametersnn

nn

n cccAA 2211,~ for large n; ...]13.3,,...669.4[ 2

21211

Scaling of The Slopes of The Transverse SM ,n(A, c)

...)601.1(**, n

nn

c

nn

DD

cS

2211

0

,

*

* *

~

q(period) = 2n

)2,( 21

11

Hyperchaotic Attractors near The Zero-Coupling Critical Point

078.0

087.0

2

1

039.0

046.0

2

1

019.0

022.0

2

1

)004.0,0065.0(

,

cA

ccAAA

2

2

cc

AAA

cc

AAA

c,1 =

12

B. Scaling Behavior near The Critical Line Segments

Consider the leftmost critical line segment with both ends cL and cR on the A = A line

Governed by the 2nd fixed point GII (x)= [f * (x)-1] with one relevant CE = 2.CTSM: = 1

*

*

Scaling of The Nonlinearity and Coupling Parameters

nn

nn cAA

~,~ for large n; )]2(,...669.4[

Scaling of The Slopes of The transverse SM ,n(A, c)

1*, n

n

ccc

nn

RLc

S

~or

,

( = 2)

cL

(= 1.457 727 …)cR

(= 1.013 402 …)

12 ’

At both ends,

(1) Scaling Behavior near The Both Ends

q(period) = 2n

13

(2) Scaling Behavior inside The Critical Line

Governed by the 3rd fixed point GIII (x)= f * (x) with no relevant CE’s and = 0*12 ’ *

Scaling Behavior: Same as that for the 1D case [Det = 12 = 0 1D]

Transverse Lyapunov exponents near the leftmost critical line segment

Inside the critical line, Synchronous Feigenbaum Attractor with < 0 on the diagonal

1D-like Scaling Behavior

When crossing both ends, Synchronous Feigenbaum State: Transversely unstable ( > 0)

23.1

,

c

AA

14

Synchronous Chaotic Attractors near The Left End of The Leftmost Critical Line

051.0

032.0||

024.0

016.0||

012.0

008.0||

)01.0,001.0(

,

cA

cccAAA L

2

2

2ccc

AAA

L

2 ccc

AAA

L

c = 2

15

2. Dissipatively-coupled case with g(x, y) = c(y2 x2)

One critical line with both endsc0 = 0 and c0 = A on the A = A line’

Stability Diagram for The Synchronous Orbits

c0c0’

16

A. Scaling Behavior near Both Ends c0 and c0

Governed by the 1st fixed point GI = 0 with two relevant CE’s 1 = (-2.502 …) and 2=2.*

xygxGxy

2][

(no constant term)

There is no component in the direction of with c = 1

Only 2 becomes a relevant one!

Scaling of The Nonlinearity and Coupling Parametersn

nn

n cAA ~,~ for large n

]2,...669.4[ 2

Scaling of The Slopes of The transverse SM ,n(A, c)

...)601.1(**, n

n

ccc

nn

RLc

S 2

or

, ~

At both ends,

B. Scaling Behavior inside The Critical Line

Governed by the 3rd fixed point GIII (x)= f * (x) with no relevant CE’s and = 0* 12

’ *

(The scaling behavior is the same as that for the 1D case.)

’

)(*1 x )]()([ *

11*12 xx

(2 = 2)

q(period) = 2n

17

Hyperchaotic Attractors near The Zero-Coupling Critical Point

172.0

193.0

2

1

086.0

096.0

2

1

043.0

048.0

2

1

)02.0,03.0(

,

cA

ccAAA

2

2

2cc

AAA

2/ cc

AAA

18

Period Doublings in Coupled Parametrically Forced Pendulums

Parametrically Forced Pendulum (PFP)

Normalized Eq. of Motion:

xtAx

txxfx

2sin)2cos(22

),,(2

Symmetrically Coupled PFPs

),(),,(,),(),,( 1222221111 xxgtxxfxxxgtxxfx

:)(2

),( 1221 xxc

xxg coupling function

)cos()( tth

O

S

l

m

= 0: Normal Stationary State = : Inverted Stationary State Dynamic Stabilization Inverted Pendulum (Kapitza)

19

Stability Diagram of The Synchronous Orbits

Same structure as in the coupled 1D maps

Critical set = zero-coupling critical point + an infinity of critical lines

Same critical behaviors as those of the coupled 1D maps

5.0,2.0

20

Scaling Behaviors near The Zero-Coupling Critical Point

007.0

011.0

2

1

004.0

006.0

2

1

002.0

003.0

2

1

)0016.0,102

,126091.0

,...709357.0(

,

6

*

*1

*1

cA

x

A

ccAAA

2

2*1

cc

AAA

cc

AAA

*1

c,1 =

21

Scaling Behaviors near The Right End of The Rightmost Critical Line

)005.0,101

,...477484.0(

,

5

*1

cA

c

cccAAA

R

R

020.0

022.0||

009.0

011.0||

004.0

006.0||

2

2*1

2ccc

AAA

R

2

*1

ccc

AAA

R

= 2

22

Summary Three Kinds of Universal Critical Behaviors Governed by the Three Fixed Points of the Reduced RG Operator (Reduced RG method: useful tool for analyzing the critical behaviors) RG results: Confirmed both in coupled 1D maps and in coupled oscillators.

[ S.-Y. Kim and H. Kook, Phys. Rev. E 46, R4467 (1992); Phys. Lett. A 178, 258 (1993); Phys. Rev. E 48, 785 (1993). S.-Y. Kim and K. Lee, Phys. Rev E 54, 1237 (1996). S.-Y. Kim and B. Hu, Phys. Rev. E 58, 7231 (1998). ]

Remarks on other relevant works 1. Extension to the even maximum-order case f (x) = 1 – A x z (z = 2, 4, 6, …) The relevant CE’s of GI (x) = 0 vary depending on z [ S.-Y. Kim, Phys. Rev. E 49, 1745 (1994). ]

2. Extension to arbitrary period p-tuplings (p = 2, 3, 4, …) cases (e.g. period triplings, period quadruplings)

Three fixed points for even p; Five fixed points for odd p [ S.-Y. Kim, Phys. Rev. E 52, 1206 (1995); Phys. Rev. E 54, 3393 (1996). ]

3. Intermittency in coupled 1D maps [ S.-Y. Kim, Phys. Rev. E 59, 2887 (1999). Int. J. Mod. Phys. B 13, 283 (1999). ]

4. Quasiperiodicity in coupled circle maps (unpublished)

*

23

Effect of Asymmetry on The Scaling Behavior

),()(

),()1()(:

1

1

tttt

tttt

xygyfy

yxgxfxT

: asymmetry parameter, 0 1

= 0: symmetric coupling 0: asymmetric coupling, = 1: unidirectional coupling

Pitchfork Bifurcation ( = 0) Transcritical Bifurcation ( 0)

Structure of The Phase Diagram and Scaling Behavior for all Same as those for = 0