1 partial synchronization in coupled chaotic systems cooperative behaviors in three coupled maps...
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Partial Synchronization in Coupled Chaotic Systems
Cooperative Behaviors in Three Coupled Maps
Sang-Yoon KimDepartment of PhysicsKangwon National University
Fully Synchronized Attractor for the Case of Strong Coupling
Breakdown of the Full Synchronization via a Blowout Bifurcation
Partial Synchronization (PS) Complete Desynchronization
)( 321 XXX
),( 3221 XXXX )( 321 XXX : Clustering
2
N Globally Coupled 1D Maps
)...,,1(1)(,))(())(()1()1( 2
1
NiaxxftxfN
ctxfctx
N
jjii
Reduced Map Governing the Dynamics of a Three-Cluster State
3)()(121
XtxtxNNN ii
Three-Cluster State
Three Coupled Logistic Maps (Representative Model)
(Each 1D map is coupled to all the other ones with equal strength.)
1)()(11
XtxtxNii
2)()(2111
XtxtxNNN ii
1st Cluster (N1) 2nd Cluster (N2) 3rd Cluster (N3)
3
)3,2,1(,))(())(()1()1(3
1
itXfpctXfctXj
jjii
pi (=Ni/N): “coupling weight factor” corresponding to the fraction of
the total population in the ith cluster
3
11
j jp
Reduced 3D Map Globally Coupled Maps with Different Coupling Weight
Investigation of the PS along a path connecting the symmetric and unidirectional coupling cases: p2=p3=p, p1=1-2p (0 p 1/3)
p1=p2=p3=1/3 Symmetric Coupling Case No Occurrence of the PSp1=1 and p2=p3=0 Unidirectional Coupling Case Occurrence of the PS
4
Transverse Stability of the Fully Synchronized Attractor (FSA)
• Longitudinal Lyapunov Exponent of the FSA
M
tt
MXf
M 1
*|| |)('|ln
1lim
• Transverse Lyapunov Exponent of the FSA
For c>c* (=0.4398), <0 FSA on the Main Diagonal
Occurrence of the Blowout Bifurcation for c=c*
FSA: Transversely Unstable (>0) for c<c*
Appearance of a New Asynchronous Attractor
Transverse Lyapunov Exponent
a=1.95
2)tymultiplici(|1|ln || c
a=1.95, c=0.5 a=1.95, c=0.5
*
321 )()()(
tX
tXtXtX
5
Type of Asynchronous Attractors Born via a Blowout Bifurcation
Unidirectional Coupling Case (p=0)Two-Cluster State: Transversely Stable Partially Synchronized Attractor on the 23 Plane Occurrence of the PS
010.0~
,021.0~
,539.0~
42.0,95.1
3
2
1
ca
Symmetric Coupling Case (p=1/3)
014.0~
,014.0~
,579.0~
42.0,95.1
3
2
1
ca
Appearance of an Intermittent Two-Cluster State on the Invariant 23 Plane ({(X1, X2, X3) | X2=X3}) through a Blowout Bifurcation of the FSA
Two-Cluster State: Transversely Unstable Completely Desynchronized (Hyperchaotic) Attractor Filling a 3D Subspace (containing the main diagonal) Occurrence of the Complete Desynchronization
6
Two-Cluster States on the 23 Plane
*32
*1 )()(,)( tt YtXtXXtX
)].()([)21()(
)],()([2)(****
1
****1
tttt
tttt
YfXfcpYfY
XfYfpcXfX
2,
2
**** YXV
YXU
.)1(2,)41(2)(1 122
1 tttttttt VUcaVVUpacVUaU
Reduced 2D Map Governing the Dynamics of a Two-Cluster State
For numerical accuracy, we introduce new coordinates:
Two-Cluster State:
003.0c003.0c
Unidirectional Coupling Case Symmetric Coupling Case
(0 p 1/3)
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Threshold Value p* ( 0.146) s.t.• 0p<p* Two-Cluster State: Transversely Stable (<0) Occurrence of the PS
• p*<p1/3 Two-Cluster State: Transversely Unstable (>0) Occurrence of the Complete Desynchronization
~
Transverse Stability of Two-Cluster States
0p
3/1p
146.0p
95.1a
M
ttt
MVUf
Mc
1
|)('|ln1
lim|1|ln
Transverse Lyapunov Exponent of the Two-Cluster State
(c cc*)
8
Mechanism for the Occurrence of the Partial Synchronization
Intermittent Two-Cluster State Born via a Blowout Bifurcation
Decomposition of the Transverse Lyapunov Exponent of the Two-Cluster State
)()(
)( blbl
bl
:),( bliL
Li
i Fraction of the Time Spent in the i Component (Li: Time Spent in the i Component)
Transverse Lyapunov Exponent of the i Component(primed summation is performed in each i component)
: Weighted Transverse Lyapunov Exponent for the Laminar (Bursting) Component
:|)(')1(|ln'1
state
it
ttii VUfc
L
)0( || 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 transverse Lyapunov exponent:
1
0
|)(')1(|ln1 L
ttt VUfc
L
d (t)
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Threshold Value p* ( 0.146) s.t. :0~~|| bl
bl ||0p<p *
p*<p1/3
Two-Cluster State: Transversely Stable Occurrence of the PS
Sign of : Determined via the Competition of the Laminar and Bursting Components
bl ||
Two-Cluster State: Transversely Unstable Occurrence of the Complete Desynchronization
(: p=0, : p=0.146, : p=1/3)
Competition between the Laminar and Bursting Components
Laminar Component
Bursting Componentpp l
lll
l oftly independen same, Nearly the :)( oft independenNearly :and
ppp bb
bbb increasingh Larger wit :)( increasingh Larger wit :, oft independenNearly :
~
|)|( lb
a=1.95, d*=10-4
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Effect of Parameter Mismatching on Partial Synchronization Three Unidirectionally Coupled Nonidentical Logistic Maps
)].,(),([),(
)],,(),([),(
),,(
3)3(
1)1(
3)3()3(
1
2)2(
1)1(
2)2()2(
1
1)1()1(
1
axfaxfcaxfx
axfaxfcaxfx
axfx
tttt
tttt
tt
: and
,,
32
33221
aaaaaa
Effect of Parameter Mismatching Partially Synchronized Attractor on the 23 ({(x(1), x(2), x(3)) | x(2)=x(3)}) Plane in the Ideal Case without Mismatching (2 = 3 = 0)
Attractor Bubbling (Persistent Intermittent Bursting from the 23 Plane)
a=1.95, c=0.42, 2=0.001, 3=0
a=1.95, c=0.42
a=1.95, c=0.42, 2=0, 3=0.001
p2=p3=p, p1=1-2p (0 p 1/3)
Reduced 2D Map Governing the Dynamics of a Two-Cluster State
mismatching parameters
),( )3()2()1(ttttt YxxXx
)].()([)(
),(
1
1
tttt
tt
YfXfcYfY
XfX
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Distribution of Local Transverse Lyapunov Exponents Probability Distribution P of Local M-time Transverse Lyapunov Exponents
Fraction of Positive Local Lyapunov Exponents
Significant Positive Tail which does not Vanish Even for Large M Parameter Sensitivity
MdPF MM
M ~)(0
• A Typical Trajectory Has Segments of Arbitrarily Long M with Positive Local Transverse Lyapunov Exponents (due to the local transverse repulsion of unstable orbits embedded in the partially synchronized attractor)
Parameter Sensitivity of the Partially Synchronized Attractor
Power-Law Decay
a=1.95, c=0.42
a=1.95c=0.42
a=1.95
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Characterization of the Parameter Sensitivity of a Partially Synchronized Attractor
Characterization of Parameter Sensitivity
• Measured by Calculating a Derivative of the Transverse Variable Denoting the Deviation from the 23 Plane with Respect to 2 along a Partially Synchronous Trajectory
Representative Value (by Taking the Minimum Value of N(X0,Y0) in an Ensemble of Randomly Chosen Initial Orbit Points)
),(min 00),( 00
YXNYX
N Parameter Sensitivity Function:
N ~ N: Unbounded Parameter Sensitivity
: Parameter Sensitivity Exponent (PSE) Used to Measure the Degree of Parameter Sensitivity
.),(),()1()(1
1*0
022
N
kkakkkNN
N aYfYXRcxSu
)exp(),()1(
1
0
MM
iiYM MaYfcR
Exponent Lyapunov Transverse time)-(M Local :M
MultiplierStability Transverse time)-(M Local :),( mmM YXR
0]for StateCluster -Two :),[( 2 kk YX
Boundedness of SN
|),(|max),( 000
00 YXSYX kNk
N
Looking only at the Maximum Values of |SN|:
]),(,2),([ 2YaYfaYaYf aY
)3()2(ttt xxu
Intermittent Behavior
a=1.95c=0.42
a=1.95c=0.42
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Characterization of the Bubbling Attractor Parameter Sensitivity Exponents (PSEs) of the Partially Synchronized Attractor on the 23
Plane
Scaling for the Average Characteristic Time
~~ *uNN
() =1/ ()
~ 1/
Average Laminar Length (i.e., average time spending near the 23 plane) of the Bubbling Attractor: ~ -
Reciprocal Relation between the Scaling Exponent and the PSE
State Bursting||
StateLaminar value)threshold(||*
*
bn
bn
uu
uu
c1*(=0.4398) > c > 0.372: Increase of
0.372 > c > 0.351 (decreasing part of ): Decrease of 0.351 > c > c2
*(=0.3376): Increase of
Increase of More Sensitive with Respect to the Parameter Mismatching
a=1.95
Partially Synchronized Attractor Bubbling Attractor (in the Presence of Parameter Mismatching)
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Effect of Noise on the Partially Synchronized Attractor
Characterization of the Noise Sensitivity of the Partially Synchronized Attractor (2=0.0005, 1=3=0)
;)(),(),(1
200
022
N
kkkkNN
N tYXRYXSu
NYXSnNnYX
N ~|),(|maxmin 000),( 00
Three Unidirectionally Coupled Noisy 1D Maps
Exponent Lyapunov Transverse time)-(M Local:M
2: Bounded Noise → Boundedness of SN: Determined by RM (same as in the parameter mismatching case)
Noise Sensitivity Exponent() = PSE() Noise Effect = Parameter Mismatching Effect
Characterization of the Bubbling Attractor
~ - ; () =1/ ()
Bubbling Attractor for a=1.95 and c=0.42
(: average time spending near the diagonal)
)exp(),( MmmM MYXR
Strength Noise :)3,2,1( ii
nceunit varia a andmean zero a with variablerandom Uniform:i
.)]()([)(
,)]()([)(
,)(
33)3()1()3()3(
1
22)2()1()2()2(
1
11)1()1(
1
tttt
tttt
tt
xfxfcxfx
xfxfcxfx
xfx
15
Partial Synchronization in Three Coupled Pendula Three Coupled Pendula
Transverse Stability of Two-Cluster States on the 23 Plane Born via a Blowout Bifurcation of the FSA
|| 1,1,1,lb
Threshold Value p* (~0.17) s.t. :0~~|| 1,1,1, bl
bl1,1, || 0p<p *
p*<p1/3
Two-Cluster State: Transversely Stable Occurrence of the PS
bl1,1, ||
Two-Cluster State: Transversely Unstable Occurrence of the Complete Desynchronization
(: p=0, : p=0.17, : p=1/3)
.21,),3,2,1(2sin)2cos(22),,(
,),,(,
1322
3
1
3
1
pppppixtAytyxf
YpccYtYXfYXpccXYXj
jjiiiij
jjiii
=1, =0.5, A=0.85 d*=10-4
Component (Bursting)Laminar for the
Exponent Lyapunov TransverseLargest Weighted:)(1,
bl
)648.0,( ** cccc
16
Unidirectional Coupling Case (p=0)
Two-Cluster State: Transversely Stable Occurrence of the PS
Symmetric Coupling Case (p=1/3)
Two-Cluster State: Transversely Unstable Occurrence of the Complete Desynchronization
=1, =0.5, A=0.85, c=0.63
1~0.648, 2~0.013,3~0.013, 4~3.790,5~4.388, 6~4.388
=1, =0.5, A=0.85, c=0.63
1~0.626, 2~0.015,3~0.013, 4~3.794,5~4.390, 6~4.415
17
Effect of Parameter Mismatching on Partial Synchronization in Three Coupled Pendula
Three Unidirectionally Coupled Nonidentical Pendula
Effect of Parameter Mismatching
Attractor Bubbling (Persistent Intermittent Bursting from the 23 Plane)
).(),,,(),(
),(),,,(),(
),,,,(,
3133333133
2122222122
111121
yycAtyxfyxxcyx
yycAtyxfyxxcyx
Atyxfyyx
parameters gmismatchin: and
,,
21,,2sin)2cos(22),,,(
32
33221
1322
AAAAAA
pppppxtAyAtyxf
=1, =0.5, A=0.85, c=0.6, 2=0.001, and 3=0
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Characterization of the Parameter Sensitivity of a Partially Synchronized Attractor
Parameter Sensitivity of a Partially Synchronized Attractor Characterized by Differentiating the Transverse Variable Denoting the Deviation from the 23 Subspace with Respect to 2 at a Discrete Time t=n.
nt
nyn
xnn
uSSS
02
)()(
2
)],([
)(min *0
)()(
*0
xiN
x
iN Parameter Sensitivity Function:
(: Parameter Sensitivity Exponent)
: Used to Measure the Degree of Parameter Sensitivity
Characterization of the Bubbling Attractor
~~ *uNN
() =1/ ()
Average Laminar Length (Interburst Interval) of the Bubbling Attractor: ~ -
Reciprocal Relation between the Scaling Exponent and the PSE
State Bursting||
StateLaminar value)threshold(||*
*
bn
bn
uu
uu
),,(),()(),()( 32 iiiyx yxztztzuutu
NiN ~)(
~ 1/
A=0.85
A=0.85
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Mechanism for the Occurrence of the Partial Synchronization in Coupled 1D Maps
Sign of the Transverse Lyapunov Exponent of the Two-Cluster State Born via a Blowout Bifurcation of the FSA: Determined via the Competition of the Laminar and Bursting Components
Summary
Similar Results: Found in High-Dimensional Invertible Period-Doubling Systems such as Coupled Parametrically Forced Pendula
)0(|| bl Occurrence of the PS
)0(|| bl Occurrence of the Complete Desynchronization
|]|[ lb
Effect of the Parameter Mismatching and Noise on the Partial Synchronization
Characterized in terms of the PSE and NSE Reciprocal Relation between the Scaling Exponent for the Average Laminar Length and the PSE(NSE) (=1/)