1

Mechanism for Occurrence of Asynchronous Hyperchaos and Chaos via Blowout Bifurcations

Dynamical Origin for the Occurrence of Asynchronous Hyperchaos and Chaos via Blowout Bifurcations

Sang-Yoon KimDepartment of PhysicsKangwon National University

Fully Synchronized Attractor for the Case of Strong Coupling

Breakup of the Chaos Synchronization via a Blowout Bifurcation

Asynchronous Hyperchaotic Attractorwith Two Positive Lyapunov Exponents

Asynchronous Chaotic Attractorwith One Positive Lyapunov Exponent

)( yx

2

N Globally Coupled 1D Maps

)...,,1(1)(,))(())(()1()1( 2

1

NiaxxftxfN

txfctxN

jjii

Reduced Map Governing the Dynamics of a Two-Cluster State

)],()([)1()()],()([)( 11 tttttttt yfxfpyfyxfyfpxfx

.)()(,)()(1111 tiitii ytxtxxtxtx

NNN

p (N2/N): “coupling weight factor” corresponding to the fraction of the total population in the 2nd cluster

Two-Cluster State

Two Coupled Logistic Maps (Representative Model)

Reduced 2D Map Globally Coupled Maps with Different Coupling Weight

Investigation of the Consequence of the Blowout Bifurcation by varying from 0 to 1.

=0 Symmetric Coupling Case Occurrence of Asynchronous Hyperchaos=1 Unidirectional Coupling Case Occurrence of Asynchronous Chaos

cp )2( and )2/()1(

.10)],()([)()],()([)1()(: 11 tttttttt yfxfcyfyxfyfcxfxT

3

Transverse Stability of the Synchronous Chaotic Attractor (SCA)

• Longitudinal Lyapunov Exponent of the SCA

N

tt

Nxf

N 1

*|| |)('|ln

1lim

• Transverse Lyapunov Exponent of the SCA

For s>s* (=0.2299), <0 SCA on the Diagonal

Occurrence of the Blowout Bifurcation for s=s*

• SCA: Transversely Unstable (>0) for s<s*

• Appearance of a New Asynchronous Attractor

Transverse Lyapunov exponent

a=1.97

|21|ln|| s

a=1.97, s=0.23

parameter coupling scaled:)2/1( cs One-Band SCA on the Invariant Diagonal

4

Type of Asynchronous Attractors Born via a Blowout Bifurcation New Coordinates

2,

2

yxv

yxu

For the accuracy of numerical calculations, we introduce new coordinates:

.])2(1[2,2)(1: 122

1 tttttttt vucavvcuavuauT

Appearance of an Asynchronous Attractor through a Blowout Bifurcation of the SCA The Type of an Asynchronous Attractor is Determined by the Sign of its 2nd Lyapunov Exponent 2 (2 > 0 Hyperchaos, 2 < 0 Chaos)

[ In the system of u and v, we can follow a trajectory until its length L becomes sufficiently long (e.g. L=108) for the calculation of the Lyapunov exponents of an asynchronous attractor.]

SCA on the invariant v=0 line Transverse Lyapunov exponent of the SCA

a=1.97, s=0.23 a=1.97

5

Evolution of a Set of Two Orthonormal Tangent Vectors under the LinearizedMap Mn [DT(zn), zn (un,vn)].

},{ )2()1(nn ww

• Reorthonormalization by the Gram-Schmidt Reorthonormalization Method

.||,,,

|,|,

)2(1

)2(1

)1(1

)1(1

)2()2()2(1)2(

1

)2(1)2(

1

)1()1(1)1(

1

)1()1(1

nnnnnnnnnn

nn

nnnn

nnn

dMMd

Mdd

M

uwwwwuu

w

ww

w

]ln[1

,1 )2,1(

1)2,1(

1

0

)2(2

1

0

)1(1

nn

L

nn

L

nn drr

Lr

L

(Direction of the 1st Vector: Unchanged)

( Has Only the Component Orthogonal to ))2(1nw )1(

1nw

• 1st and 2nd Lyapunov Exponents 1 and 2

)1(nw

)2(nw

)1(nnM w

)2(nnM w

)1(1nw

)2(1nw

nM

nz 1nz1nM1nM

Computation of the Lyapunov Exponents 1 and 2 for a Trajectory Segment with Length L

6

Second Lyapunov Exponent of the Asynchronous Attractor

Threshold Value * ( ~ 0.852) s.t.• < * Asynchronous Hyperchaotic Attractor (HCA) with 2 > 0

• > * Asynchronous Chaotic Attractor (CA) with 2 < 0

(dashed line: transverse Lyapunov exponent of the SCA)

HCA for = 0 CA for = 1

a=1.97, L=108

a = 1.97s = 0.00161 = 0.61572 = 0.0028

a = 1.97s = 0.00161 = 0.60872 = 0.0024

(: =0, : =0.852, : =1)

7

Mechanism for the Occurrence of Asynchronous Hyperchaos and Chaos

Intermittent Asynchronous Attractor Born via a Blowout Bifurcation

Decomposition of the 2nd Lyapunov Exponent 2 of the Asynchronous Attractor

)(2)(

)(2

blbl

bl

:),( bliL

Li

i Fraction of the Time Spent in the i Component (Li: Time Spent in the i Component)

2nd Lyapunov Exponent of the i Component(primed summation is performed in each i component)

: Weighted 2nd Lyapunov Exponent for the Laminar (Bursting) Component

)0( || 222222 llbbl

d = |v|: Transverse Variabled*: Threshold Value s.t. d < d*: Laminar Component (Off State), d > d*: Bursting Component (On State).

We numerically follow a trajectory segment with large length L (=108), and calculate its 2nd Lyapunov exponent.

d (t)

:1

state

)2(2

in

nii r

L ’

8

Threshold Value * (~ 0.852) s.t. :0~~|| 222 bl

bl22 || < *

> *

Asynchronous Hyperchaotic Attractor with 2 > 0

Sign of 2 : Determined via the Competition of the Laminar and Bursting Components

bl22 ||

Asynchronous Chaotic Attractor with 2 < 0

(: =0, : =0.852, : =1)

Competition between the Laminar and Bursting Components

Laminar Component

Bursting Component oftly independen same, Nearly the :)( oft independenNearly :and 222

ll

lll

increasingth Smaller wi :)( increasingth Smaller wi :, oft independenNearly : 222b

bbb

b

|)|( 22lb

a=1.97, d*=10-5

9

• : Dependent on d *

As d * Decreases, a Fraction of the Old Laminar Component is Transferred to the New Bursting Component:

• 2 Depends Only on the Difference Between the Strength of the Laminar and Bursting Components. The Conclusion as to the Type of Asynchronous Attractors is Independent of d *.

)(2

bl

)( and )(|| 22 bl

Effect of the Threshold Value d * on )(2

bl

In the limit d *0,

a=1.97

.|)|( and 0|| 22222 lbl

(: d*=10-6, : d*=10-8, : d*=10-10)

|)|( 22lb

10

System: Coupled Hénon Maps

,)],()([)(

,)],()([)1()()2()2(

1)2()1()2()2()2(

1

)1()1(1

)1()2()1()1()1(1

ttttttt

ttttttt

bxyxfxfcyxfx

bxyxfxfcyxfx

.1 2axxf

• Type of Asynchronous Attractors Born via Blowout Bifurcations|| 222

lb

Threshold Value * ( 0.905) s.t. 0~||~ 222 lb~

(s*=0.787 for b=0.1 and a=1.83)

d *=10-4 d *=10-4L=108

2/|)||(| )2()1( vvd

Blowout Bifurcations in High Dimensional Invertible Systems

.,])2(1[2

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

1)2()1()1()1(

1

)1()2(1

)2()1()1(2)1(2)1()1(1

tttttt

tttttttt

bvvvvucav

buuuvcuavuau

.2

,2

,2

,2

)2()1()2(

)2()1()1(

)2()1()2(

)2()1()1( yy

vxx

vyy

uxx

u

New Coordinates:

(: =0, : =0.905, : =1)

For < * HCA with 2 > 0, for > * CA with 2 < 0.

11

HCA for = 0 CA for = 1

a=1.83, s=-0.00161 0.43402 0.0031

~~

System: Coupled Parametrically Forced Pendulums

),(),,(),(

),()1(),,(),()1(

212222122

121111211

yyctyxfyxxcyx

yyctyxfyxxcyx

.2

,2

,2

,2

212

211

212

211

yyv

xxv

yyu

xxu

a=1.83, s=-0.00161 0.44062 -0.0024

~~

New Coordinates:

.)2(2sin2cos)2cos(22

,)2(

,2cos2sin)2cos(22

,

2112

22

121

2112

22

121

cvvutAvv

cvvv

cvvutAuu

cvuu

.2sin)2cos(22),,( 2 xtAxtxxf cs )2/1(

12

Threshold Value * ( 0.84) s.t. 0~||~ 222 lb~

HCA for = 0 CA for = 1

1 0.6282 0.017

~~

1 0.6482 -0.008

~~

A=0.85s =-0.006

A=0.85s=-0.005

L=107d *=10-4 d *=10-4

|| 222lb

• Type of Asynchronous Attractors Born via Blowout Bifurcations(s*=0.324 for =1.0, =0.5, and A=0.85)

2/|)||(| 21 vvd

(: =0, : =0.84, : =1)

For < * HCA with 2 > 0, for > * CA with 2 < 0.

13

Mechanism for the Occurrence of the Hyperchaos and Chaos via Blowout Bifurcations

Sign of the 2nd Lyapunov Exponent of the Asynchronous Attractor Born via a Blowout Bifurcation of the SCA: Determined via the Competition of the Laminar and Bursting Components

Summary

Similar Results: Found in High-Dimensional Invertible Period-Doubling Systems such as Coupled Hénon Maps and Coupled Parametrically Forced Pendula

)0(|| 222 bl Occurrence of the Hyperchaos

|]|[ 222lb

)0(|| 222 bl Occurrence of the Chaos

14

Effect of Asynchronous UPOs on the Bursting Component

Change in the Number of Asynchronous UPOs with respect to s (from the first transverse bifurcation point st to the blow-out bifurcation point s*)

• Symmetric Coupling Case (=0)

• Transverse PFB of a Synchronous Saddle • Asynchronous PDB

q

q

q

q

2q

Type of Bifs.

No. of Bifs.

No. of Saddles(Ns)

No. of Repellers(Nr)

TransversePFB 12 +24 0

Asyn. PDB 16 -16 +16

Total No. of UPOs +8 +16

(Period q=11)

15

• Unidirectional Coupling Case (=1)

• Asynchronous SNB • Asynchronous PDB

Type of Bifs.

No. of Bifs.

No. of Saddles(Ns)

No. of Repellers(Nr)

Asyn. SNB 21 +21 +21

Transverse TB

12 +12 -12

Asyn. PDB 9 -9 +9

Total No. of UPOs +24 +18

q

2q

• Transverse TB

q

q

q

q

(Period q=11)

16

Change in the Number of Asynchronous UPOs at the Blow-Out Bifurcation Point s* (=0.190) with respect to

Type of Bifs.

No. of Bifs.

Increased No. of Saddles

Increased No. of Repellers

SNB 13 +13 +13

Reverse SNB

4 -4 -4

PDB 10 -10 +10

Reverse PDB

17 +17 -17

Total Increased

No. of UPOs+16 +2

• SNB • Reverse SNB • PDB • Reverse PDB

q

q

q

q

q

2q

q

2q

(Period q=11, Ns: No. of Saddles, Nr: No. of Repellers)

17

Transition from Chaos to Hyperchaos

For s = s* ( 0.163), a Transition from Chaos to Hyperchaos Occurs.~

1 0.4782 0.018

~~

a=1.83s=0.155=1

18

Characterization of the On-Off Intermittent Attractors Born via Blow-Out Bifurcations

d: Transverse Variable (Denoting the Deviation from the Diagonal) d < d *: Laminar State (Off State) d d *: Bursting State (On State)

• Distribution of the Laminar Length:

• Scaling of the Average Laminar Length:

• Scaling of the Average Bursting Amplitude:

,~)(*/2/3 eP

*~ ppd

1*)(~ pp

2** )( pp

p=p*: Blow-Out Bifurcation Point

19

Phase Diagrams in Coupled 1D Maps System: Coupled 1D Maps:

),,(

),,()1(:

1

1

tttt

tttt

xygcyfy

yxgcxfxT

.1 2axxf

Dissipative Coupling Case with g(x, y) = f(y) – f(x)

• Periodic Synchronization

Symmetric Coupling (=0) Unidirectional Coupling (=1)

Horizontal Lines: Longitudinal Bifurcations Synchronous Period-Doubling Bifurcations, Nonhorizontal Solid and Dashed Lines: Transverse Bifurcations (Solid Lines: Period-Doubling Bifurcations. Dashed Lines for =0 and 1: Pitchfork and Transcritical Bifurcations, Respectively.)

20

• Chaotic Synchronization

Symmetric Coupling (=0) Unidirectional Coupling (=1)

Hatched Region: Strong Synchronization, Light Gray Region: Bubbling,Dark Gray Region: Riddling Solid or Dashed Lines: First Transverse Bifurcation Lines (Solid Lines: Period-Doubling Bifurcations. Dashed Lines for =0 and 1: Pitchfork and Transcritical Bifurcations, Respectively.)Solid Circles: Blow-Out Bifurcation

21

Inertial Coupling Case with g(x, y) = y – x

• Periodic Synchronization

Symmetric Coupling (=0) Unidirectional Coupling (=1)

Horizontal Lines: Longitudinal Bifurcations Synchronous Period-Doubling Bifurcations, Nonhorizontal Solid and Dashed Lines: Transverse Bifurcations (Solid Lines: Period-Doubling Bifurcations. Dashed Lines for =0 and 1: Pitchfork and Transcritical Bifurcations, Respectively.)

22

• Chaotic Synchronization

Symmetric Coupling (=0) Unidirectional Coupling (=1)

Hatched Region: Strong Synchronization, Light Gray Region: Bubbling,Dark Gray Region: Riddling Solid or Dashed Lines: First Transverse Bifurcation Lines (Solid Lines: Period-Doubling Bifurcations. Dashed Lines for =0 and 1: Pitchfork and Transcritical Bifurcations, Respectively.)Solid Circles: Blow-Out Bifurcation