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
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• : 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
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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.
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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(
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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.
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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
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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)
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• 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)
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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)
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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
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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.)
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• 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