quasiperiodic dynamics in coupled period-doubling systems
DESCRIPTION
Quasiperiodic Dynamics in Coupled Period-Doubling Systems. Sang-Yoon Kim Department of Physics Kangwon National University Korea. Nonlinear Systems with Two Competing Frequencies. Mode Lockings, Quasiperiodicity, and Chaos. Symmetrically Coupled Period-Doubling Systems - PowerPoint PPT PresentationTRANSCRIPT
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Quasiperiodic Dynamics in Coupled Period-Doubling Systems
Sang-Yoon Kim
Department of Physics
Kangwon National University
Korea
Nonlinear Systems with Two Competing Frequencies
Mode Lockings, Quasiperiodicity, and Chaos
2
Symmetrically Coupled Period-Doubling Systems
Building Blocks: Period-Doubling Systems such as the 1D Map, Hénon Map, Forced Nonlinear Oscillators, and Autonomous Oscillators
Coupled Systems: Generic Occurrence of Hopf Bifurcations Quasiperiodic Transition
Purpose • Investigation of Mode Lockings, Quasiperiodicity, and Torus Doublings Associated with Hopf Bifurcations
• Comparison of the Quasiperiodic Behaviors of The Coupled Period-Doubling Systems with Those of the Circle Map (representative model for quasiperiodic systems with two competing frequencies)
3
~
~
R L
Ve= V0 sint
Rc
R L
R L
Ve= V0 sint
II
V0
L=470mH, f=3.87kHz, R=244 Single p-n junction resonator Period-doubling transition
Resistively coupled p-n junction resonators Quasiperiodic transition
L=100mH, Rc=1200, f=12.127kHz
V0
Quasiperiodic Transition in Coupled p-n Junction Resonators[ R.V. Buskirk and C. Jeffries, Phys. Rev. A 31, 3332 (1985). ]
Hopf Bifurcation
4
Quasiperiodic Transition in Coupled Parametrically Forced Pendulums (PFPs)
Single PFP
Symmetrically Coupled PFPs
Period-Doubling Transition
5.0,2.0
Quasiperiodic Transition
0.1C
.2sin)2cos(22),,(
,)(2
),,(,
,)(2
),,(,
2
2122222
1211111
xtAxtxxf
xxC
tyxfyyx
xxC
tyxfyyx
A
A
A
Hopf Bifurcation
5
Quasiperiodic Transition in Coupled Rössler Oscillators
Single Rössler Oscillator
Symmetrically Coupled Rössler Oscillators
Period-Doubling Transition
Quasiperiodic Transition
2.0ba
05.0
).()(
,,
),()(
,,
12222
222222
21111
111111
xxcxzbz
ayxyzyx
xxcxzbz
ayxyzyx
Hopf Bifurcation
6
Hopf Bifurcations in Coupled 1D Maps
.),(1
,),(1:
21
21
tttt
tttt
xygAyy
yxgAxxT
Two Symmetrically Coupled 1D Maps
Phase Diagram for The Linear Coupling Case with g(x, y) = C(y x)
Synchronous Periodic Orbits
Antiphase Orbits with Phase Shift ofHalf A Period (in a gray region)
Quasiperiodic Transition through A Hopf Bifurcation
Transverse PDB
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Type of Orbits in Symmetrically Coupled 1D Maps
Exchange Symmetry:
Consider an orbit {zt}:
Strongly-Symmetric Orbits ()
Synchronous orbit on the diagonal ( = 0°)
Weakly-Symmetric Orbits (with even period n)
Antiphase orbit with phase shift of half a period () ( = 180°)
Asymmetric Orbits (, ) A pair of conjugate orbits {zt} and Dual Phase Orbits
),();(1 yxzzTz tt
),(),(; xyyxSTSTS
tt zzS
22/
nttn
t zzTzS
}{ tzS
1.0,5.1 CA
Symmetrically Coupled 1D Maps
Symmetry line: y = x (Synchronization line)
(In-phase Orbits)
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Self-Similar Topography of The Antiphase Periodic Regimes
Antiphase Periodic Orbits in The Gray Regions
Self-Similarity near The Zero- Coupling Critical Point
Nonlinearity and coupling parameter scaling factors: (= 4.669 2…), (= 2.502 9…)
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Hopf Bifurcations of Antiphase Orbits
Loss of Stability of An Orbit with Even Period n through A Hopf Bifurcation when its Stability Multipliers Pass through The Unit Circle at = e2i.
}{ *tz
Birth of Orbits with Rotation No. ( : Average Rotation Rate around a mother orbit point per period n of the mother antiphase orbit)
Quasiperiodicity (Birth of Torus)
irrational numbers Invariant Torus
Mode Lockings (Birth of A Periodic Attractor)
(rational no.) r / s (coprimes) Occurrence of Anomalous Hopf Bifurcations
r: even Symmetry-Conserving Hopf Bifurcation Appearance of a pair of symmetric stable and unstable orbits of rotation no. r / s r: odd Symmetry-Breaking Hopf Bifurcation Appearance of two conjugate pairs of asymmetric stable and unstable orbits of rotation no. r / s
18.0
,24.1
C
A
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Arnold Tongues of Rotation No. (= r / s)
Unstable manifolds of saddle points flow into sinks, and thus union of sinks, saddles, and unstable manifolds forms a rational invariant circle.
A=1.266 and C= 0.196
A Pair of SymmetricSink and Saddle
Two Pairs of AsymmetricSinks and Saddles
52 83A=1.24 and C= 0.199
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Bifurcations inside Arnold’s Tongues1. Period-Doubling Bifurcations
(Similar to the case of the circle map) Case of A Symmetric Orbit
Case of An Asymmetric Orbit
Hopf Bifurcation from The Antiphase Period-2 Orbit
(e.g. see the tongue of rot. no. 28/59)
(e.g. see the tongue of rot. no. 19/40)
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2. Hopf Bifurcations Tongues inside Tongues
2nd Generation(daughter tongues inside their mother tongue of rot. no. 2/5)
3rd Generation(daughter tongues inside their mother tongue of rot. no. 4/5)
2/5
6/7
4/55/6
4/5
6/7
4/5
8/9
5/6
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Transition from Torus to Chaos Accompanied by Mode Lockings
(Gradual Fractalization of Torus Loss of Smoothness)
Smooth Torus Wrinkled Torus Fractal Torus (Strange Nonchaotic Attractor) ? Mode Lockings Chaotic Attractor
(Wrinkling behavior of torus is masked by mode lockings.)
052.0,0
1.0,349.1
21
CA
044.0,0
1.0,3525.1
21
CA
012.0
,008.0
23/6
1.0,3526.1
2
1
CA
025.0,006.0
1.0,3532.1
21
CA
~ ~ ~
~
~
~
~~
009.0998.02 D 002.0895.02 D
004.0123.12 D
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Quasiperiodic Dynamics in Coupled 1D Maps
Hopf Bifurcations of Antiphase Orbits
Quasiperiodicity (invariant torus) + Mode Lockings
Question: Coupled 1D Maps may become a representative model for the quasiperiodic behavior in symmetrically coupled system?
No !
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Torus Doublings in Symmetrically Coupled Oscillators Occurrence of Torus Doublings in Coupled Parametrically Forced Pendulums ( = 0.2, = 0.5, and A = 0.352)
48.0C 517.0C
524.0C 534.0C
DoubledTorus
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Torus Doublings in Coupled Hénon Maps
,,,1
,,,1:
1
2
1
1
2
1
ttttttt
ttttttt
buvxugAuvu
bxyuxgAxyxT
.,
10
xuCuxg
b
Symmetrically Coupled Hénon Maps
Torus doublings may occur only in the (invertible) N-D maps (N 3).
Characterization of Torus Doublings by The Spectrum of Lyapunov Exponents
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Torus Doublings for b = 0.5 and A = 2.05
29.0C 305.0C
31.0C 34.0C
reverse normal
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Damping Effect on Torus Doublings and Mode Lockings
Torus doublings occur for b > 0.3. (No torus doublings for b < 0.3)
As b is increased, the region of mode lockings decreases.
~
b = 0.7b = 0.5b = 0.3
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b 0.3 0.5 0.7
Hyperchaos 24.36 % 20.64 % 12.03 %
Chaos 37.55 % 34.86 % 19.88 %
Mode lockings 15.66 % 13.52 % 11.13 %
Torus 22.40 % 27.81 % 48.19 %
Doubled torus 0.03 % 3.17 % 8.17 %
Period-4 torus 0 % 0 % 0.60 %
Ratios of Hyperchaos, Chaos, Torus, and Mode Lockings
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Summary
In Symmetrically Coupled Period-Doubling Systems, Mode Lockings and
Quasiperiodicity occur through Hopf Bifurcations of Antiphase Orbits
(Representative model: Coupled 1D Maps).
Bifurcations inside the Arnold tongues become richer than those in the case of the
circle map
Torus Doublings also occur in Symmetrically Coupled Hénon Maps when the
damping parameter becomes larger than a threshold value, which is in contrast
to the coupled 1D maps without torus doublings.
Effect of Asymmetry on The Quasiperiodic Behavior
10,),(1
,),()1(1:
21
21
tttt
tttt
xygAyy
yxgAxxT
Threshold value *, s.t. 0 < < * Robustness of The Quasiperiodic Behavior
> * No Hopf Bifurcation (No Quasiperiodic Behavior)
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Complex Dynamics in Symmetrically Coupled Systems
In-phase orbits Universal Scalings of Period Doublings
Antiphase orbits Quasiperiodic Dynamics (Hopf Bif.)
Dual phase orbits What’s their dynamics?
Period Doublings
Feigenbaum lines
Scaling near both end?
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Stability Analysis in Coupled Hénon Maps
,,),(1
,,),(1:
12
1
12
1
ttttttt
ttttttt
buvxugAuvu
bxyuxgAxyxT
qbDMMBMADDABAP 222
21234 ,]Tr Tr [ ,Tr ;
Complex Quadruplet: *,,, *
DD Hopf Bifurcation
.)(),( xuCuxg
Consider an orbit of period q. Its stability is determined by its Stability Multipliers which are the eigenvalues of the linearized map M (=DTq) of Tq around the period-q orbit.
M: Dissipative Symplectic Map Eigenvalues come into pairs lying on the circle of radius D1/4