mechanism for occurrence of asynchronous hyperchaos and chaos via blowout bifurcations
DESCRIPTION
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 Kim Department of Physics Kangwon National University. - PowerPoint PPT PresentationTRANSCRIPT
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)
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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.
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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
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)
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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)
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)
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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.)
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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
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