sang-yoon kim department of physics kangwon national university
DESCRIPTION
Complex Dynamics in Coupled Period-Doubling Systems; Mode Locking, Quasiperiodicity, and Torus Doublings. Sang-Yoon Kim Department of Physics Kangwon National University. Nonlinear Systems with Two Competing Frequencies (e.g. circle map). Mode Lockings, Quasiperiodicity, and Chaos. - PowerPoint PPT PresentationTRANSCRIPT
1
Complex Dynamics in Coupled Period-Doubling Systems;Mode Locking, Quasiperiodicity, and Torus Doublings
Sang-Yoon Kim
Department of Physics
Kangwon National University
Nonlinear Systems with Two Competing Frequencies (e.g. circle map)
Mode Lockings, Quasiperiodicity, and Chaos
Coupled Period-Doubling Systems (e.g. coupled logistic maps)
Single System: Period-Doubling Transition to ChaosCoupled Systems: Generic Occurrence of Hopf Bifurcations Mode Locking, Quasiperiodicity, Torus Doubling, Chaos, Hyperchaos
2
~
~
R L
Ve= V0 sin(2ft)
Rc
R L
R L
Ve= V0 sin(2ft)
II
V0
L=470mH, R=244, f=3.87kHz
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
3
Quasiperiodic Transition in Coupled Pendula
Single Pendulum
Symmetrically Coupled Pendula
Period-Doubling Transition
1,7.0
Quasiperiodic Transition
8.0
.2sin)2cos(22),,(
,)(),,(,
,)(),,(,
2
2122222
1211111
xtAxtxxf
xxtyxfyyx
xxtyxfyyx
A
A
A
Hopf Bifurcation
4
Quasiperiodic Transition in Coupled Rössler Oscillators
Single Rössler Oscillator
Symmetrically Coupled Rössler Oscillators
Period-Doubling Transition
Quasiperiodic Transition
2.0ba
1.0
).(,
),(
),(,
),(
222222
12222
111111
21111
cxzbzayxy
xxzyx
cxzbzayxy
xxzyx
Hopf Bifurcation
5
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) = (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
a
6
Type of Orbits in Symmetrically Coupled 1D Maps
Exchange Symmetry: STS = T; S(x,y) = (y,x)
Consider an orbit {zt}:
Strongly-Symmetric Orbits () Szt = zt (In-phase Orbits)
Synchronous orbit on the diagonal ( = 0°)
Weakly-Symmetric Orbits (with even period n) Szt = Tn/2 zt = zt+n/2
Antiphase orbit with phase shift of half a period () ( = 180°)
Asymmetric Orbits (, ) A pair of conjugate orbits {zt} and {Szt} Dual Phase Orbits
),();(1 yxzzTz tt
01.0,31.1 a
Symmetrically Coupled 1D Maps
Symmetry line: y = x (Synchronization line)
[Periodic Orbits can be Classified in Terms of the Periods and Phase Shifts (p, q): q = 0, …, p1]
7
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…)
8
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 Standard Hopf Bifurcation Appearance of a pair of symmetric stable and unstable orbits of rotation no. r / s r: odd Nonstandard Double Hopf Bifurcation Appearance of two conjugate pairs of asymmetric stable and unstable orbits of rotation no. r / s
25.0
,2.1
a
9
Arnold Tongues of Rotation No. (= r / s)
a=1.266 and = 0.169
A Pair of SymmetricSink and Saddle
Two Conjugate Pairs of Asymmetric
Sinks and Saddles
52
83
a=1.24 and = 0.2
Standard Hopf Bifurcation
Nonstandard Double Hopf Bifurcation
10
Bifurcation Patterns 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-4 Orbit
(e.g. see the tongue of rot. no. 18/47)
(e.g. see the tongue of rot. no. 17/44)
SNB
PFB
PDB
SNB
PDB
SNB
SNB
11
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)
3. Period-Doubling and Hopf Bifurcations
HB
SNB
SNB
HB SNB
PFB
PDB
(e.g. see the tongue of rot. no. 2/5)
(e.g. see the tongue of rot. no. 12/31)
12
Transition from Torus to Chaos Accompanied by Mode Lockings
Smooth Torus Wrinkled Torus Mode Lockings Chaotic Attractor
(Wrinkling behavior of torus is masked by mode lockings.)
23.1a
24.1a
238.1a
26.1a
008.0
8/3
1
2.0
01 01
038.01
13
Effect of Asymmetry on Hopf Bifurcations
.)(1
,)()1(1:
21
21
tttt
tttt
yxcayy
xycaxxT
System
=0: symmetric coupling=1: unidirectional coupling
0<* (=0.392) : Hopf Bifurcations Leading to Quasiperiodicity and Mode Locking*<1: Period-Doubling Bifurcations
0 1
14
Hopf Bifurcations in Coupled Pendula
Standard Hopf Bifurcation Nonstandard Double Hopf Bifurcation
1,7.0
A=2.75 and = 1.11
9/2
A=2.75 and = 1.156
5/1
15
Dynamical Behaviors of Symmetrically Coupled 1D Maps
Hopf Bifurcations of Antiphase Orbits
Quasiperiodicity (invariant torus) + Mode Lockings
Another Interesting Behavior of Symmetrically Coupled Oscillators: Torus Doublings (no occurrence in coupled 1D maps)
16
Torus Doublings in Symmetrically Coupled Pendula Occurrence of Torus Doublings in Symmetrically Coupled Pendula ( = 0.1, = 1, and = 0.6)
DoubledTorus
016.0,008.0,5174.1 21 A
037.0,008.0,517.1 21 A
17
Torus Doublings in Coupled Hénon Maps
),()1(,)(),()(1)()1(
),()1(,)(),()(1)()1(:
22122222
11212111
tbxtytxtxgtaxtytx
tbxtytxtxgtaxtytxT .,
10
1221 xxxxg
b
Symmetrically Coupled Hénon Maps (Representative Model for Poincaré Maps of Coupled Period-Doubling Oscillators)
Torus doublings may occur only in the (invertible) N-D maps (N 3).
Characterization of Torus Doublings by the Spectrum of Lyapunov Exponents
18
Torus Doublings for b = 0.5 and = -0.305
024.0,0,04.2 21 a 017.0,0,05.2 21 a
01.0,007.0,07.2 21 a
27/4
031.0,049.0,08.2 21 a
19
Damping Effect on the Ratios of Dynamical Regimes
Occurrence of Torus Doublings for b > 0.3~
b = 0.7b = 0.5b = 0.2
Doubled Torus
DoubledTorus
Increase in the Ratios of the Smooth Torus(T), Doubled Torus(2T), and Quadrupled Torus(4T)
Decrease in the Ratios of the Mode Locking, Chaos, and Hyperchaos
20
Summary
Mode Lockings and Quasiperiodicity via Hopf Bifurcations of Antiphase Orbits in Coupled 1D Maps
Occurrence of Anomalous Hopf Bifurcations; Standard and Nonstandard Double Hopf Bifurcations
Occurrence of Torus Doublings in Symmetrically Coupled Hénon Maps for b > b*
(in contrast to the coupled 1D maps without torus doublings)
Effect of the Asymmetry of Coupling on Hopf Bifurcations
nsBifurcatio Doubling-Period :1
s)(RobustnesnsBifurcatio Hopf :0 s.t. valueThreshold*
**
Universality
Confirmed in Symmetrically Coupled Period-Doubling Oscillators such as the Coupled Pendula and Rössler Oscillators
21
Anomalous Hopf Bifurcations of Antiphase Orbits
Anitphase Orbits
{zt = (xt, yt)}: Antiphase Orbit with an Even Period n in Coupled 1D Maps T with the Exchange Symmetry S
ttn
tnttn
tttn zzTSzRzzTzSzzT )()()()(,)( 2/
2/2/
zt: Fixed Point of Both Tn and R
Tn=RR (R: Half-Cycle Map)
Standard Hopf Bifurcation in RStability Multiplier: = e 2i p/q Appearance of a Pair of Stable and Unstable Orbits of Rotation Number R (=p/q)
Anomalous Hopf Bifurcation in Tn
Stability Multiplier of zt in Tn: = e 2i r/s = e 2i (2p)/q
(1) Standard Hopf Bifurcation (q: odd r: even) r = 2p (even), s = q (odd) Appearance of a Pair of Stable and Unstable Orbits of Rotation Number (=r/s)
(2) Nonstandard Double Hopf Bifurcation (q: even r: odd) r = p (odd), s = q/2 Appearance of Two Pairs of Stable and Unstable Orbits of Rotation Number (=r/s)