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

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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

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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 .

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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

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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

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Intermittent Route in the Quasiperiodically Forced Ring Map

1). mod(,2sin,2cos2sin

121

21

nnna

nn

nnna

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Quasiperiodically Forced Ring Map Phase Diagram (b=0.01)

Smooth Torus Intermittent SNA

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a

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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*

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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

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yxx

Quasiperiodically Forced Toda Oscillator

Smooth Torus Intermittent SNA

Phase Diagram (=0.8, 1=2)

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a

6.7004.0

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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*

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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.