Upload
osborn-gilmore
View
231
Download
0
Embed Size (px)
DESCRIPTION
3 Ring-Shaped Unstable Set Birth of a Ring-Shaped Unstable Set (RUS) via a Phase-Dependent Saddle-Node Bifurcation RUS of Level k=7: Composed of 13 Small Rings Each Ring: Composed of Stable (Black) and Unstable (Gray) Orbits with Period F 7 (=13) (Unstable Part: Toward the Smooth Torus They may Interact.) Evolution of the Rings Appearance of CA via Period-Doubling Bifurcations (PDBs) and Its Disappearance via a Boundary Crisis (Upper Gray Line: Period-F 7 (=13) Orbits Destabilized via PDBs) Expectation: In the Quasiperiodic Limit, the RUS forms a Complicated Unstable Set Composed of Only Unstable Orbits
Citation preview
1
Universality for the Intermittent Route to Strange Nonchaotic Attractors in Quasiperiodically Forced Systems
W. Lim and S.-Y. Kim Kangwon National University
1). mod(,,2cos
1
12
1
nn
nnnnnn bxyyxax
Quasiperiodically Forced Hénon Map
.2
15
Appearance of Intermittent Strange Nonchaotic Attractor (SNA)
Smooth Torus Intermittent SNA
415.096.0
a
32.4006.0
86416.096.0
1
a
Property of SNAs: 1. No Sensitivity to Initial Condition (<0) 2. Fractal Phase Space Structure
986857416.0*
2
Phase Diagram
05.0b
Route a: IntermittencyRoute b or c: Interior crises of SNA or chaotic attractor (CA)Route d, e, or f: Boundary crises of Smooth Torus, SNA, or CA
Smooth Torus (Light Gray): T and 2TCA (Black), SNA (Gray and Dark Gray)
Rational Approximation (RA)• Investigation of the Intermittent Transition in a Sequence of Periodically Forced Systems with Rational Driving Frequencies k, Corresponding to the RA to the Quasiperiodic Forcing ( ) :
• Properties of the Quasiperiodically Forced Systems Obtained by Taking the Quasiperiodic Limit k .
1 and 0,;/ 10111 FFFFFFF kkkkkk
2/)15(
3
Ring-Shaped Unstable Set
7,3707.0,85.0 ka
7,375.0,86.0 ka
Birth of a Ring-Shaped Unstable Set (RUS) via a Phase-Dependent Saddle-Node Bifurcation
• RUS of Level k=7: Composed of 13 Small Rings
Each Ring: Composed of Stable (Black) and Unstable (Gray) Orbits with Period F7 (=13)
(Unstable Part: Toward the Smooth Torus They may Interact.)
Evolution of the Rings
• Appearance of CA via Period-Doubling Bifurcations (PDBs) and Its Disappearance via a Boundary Crisis
(Upper Gray Line: Period-F7 (=13) Orbits Destabilized via PDBs)
Expectation: In the Quasiperiodic Limit, the RUS forms a Complicated Unstable Set Composed of Only Unstable Orbits
4
Mechanism for the Intermittency
7,4.0,96.0 ka 7,4015.0,96.0 ka 7,4045.0,96.0 ka
168.01
In the RA of level k=7, phase-dependent SNB between smooth torus and RUS occurs.With further increase of , interior crisis with the RUS occurs. Appearance of gaps, filled by intermittent chaotic attractors. RA of Intermittent SNA
5
Intermittent Route in the Quasiperiodically Forced Ring Map
1). mod(,2sin,2cos2sin
121
21
nnna
nn
nnna
nn
xbyybyxxx
Quasiperiodically Forced Ring Map Phase Diagram (b=0.01)
Smooth Torus Intermittent SNA
1801.095.2
a
5.20004.0
276180.095.2
1
a
.2/)15(
Route a: IntermittencyRoute b or c: Interior crises of SNA or CASmooth Torus (Light Gray): T and 2TCA (Black), SNA (Gray and Dark Gray)
Appearance of Intermittent SNA991275180.0*
6
Mechanism for the Intermittency in the Quasiperiodically Forced Ring Map
7,177.0,95.2 ka 7,1775.0,95.2 ka 7,178.0,95.2 ka
In the RA of level k=7, phase-dependent SNB between smooth torus and RUS occurs.With further increase of , interior crisis with the RUS occurs. Appearance of gaps, filled by intermittent chaotic attractors. RA of Intermittent SNA
053.01
7
Intermittent Route in the Quasiperiodically Forced Toda Oscillator
1). mod(
),2cos2cos1(
,
21
2
1
42
taeyy
yxx
Quasiperiodically Forced Toda Oscillator
Smooth Torus Intermittent SNA
Phase Diagram (=0.8, 1=2)
764.018
a
6.7004.0
15765.018
1
a
.2/)15(
Route a: IntermittencyRoute b or c: Interior crises of SNA or CASmooth Torus (Light Gray): T and 2TCA (Black), SNA (Gray and Dark Gray)
Appearance of Intermittent SNA585139765.0*
8
Mechanism for the Intermittency in the Quasiperiodically Forced Toda Oscillator
7,715.0,18 ka 7,729.0,18 ka 7,74.0,18 ka
In the RA of level k=7, phase-dependent SNB between smooth torus and RUS occurs.With further increase of , interior crisis with the RUS occurs. Appearance of gaps, filled by intermittent chaotic attractors. RA of Intermittent SNA
102.01
9
Summary
• Appearance of Intermittent SNAs in Quasiperiodically Forced Systems
Tongue of Quasiperiodic Motion, Penetrating into the Chaotic Region, near the Terminal Point of the Torus-Doubling Bifurcation Line
When Passing the Upper Boundary of the Tongue, a Smooth Torus Transforms into and Intermittent SNA.
• Universal Mechanism for the Intermittency
Transition to the Intermittent SNA occurs via a Collision with a Ring-ShapedUnstable Set.