Universality%20for%20the%20Intermittent%20Route%20to%20Strange%20Nonchaotic%20Attractors%20in%20Quasiperiodically%20Forced%20Systems

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Universality for the Intermittent Route to
Strange Nonchaotic Attractors in
Quasiperiodically Forced Systems
W. Lim and S.-Y. Kim
Kangwon National University
? Quasiperiodically Forced Hénon Map
? Appearance of Intermittent Strange Nonchaotic
Attractor (SNA)
Smooth Torus
Intermittent SNA
Property of SNAs 1. No Sensitivity to Initial
Condition (?lt0)
2. Fractal Phase Space Structure
2
? Phase Diagram
Route a Intermittency Route 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 2T CA (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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Ring-Shaped Unstable Set

• Birth of a Ring-Shaped Unstable Set (RUS) via a
  Phase-Dependent Saddle-Node Bifurcation

RUS of Level k7 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

• In the RA of level k7, 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
Phase Diagram (b0.01)
? Quasiperiodically Forced Ring Map
Route a Intermittency Route b or c Interior
crises of SNA or CA Smooth Torus (Light Gray) T
and 2T CA (Black), SNA (Gray and Dark Gray)
? Appearance of Intermittent SNA
Smooth Torus
Intermittent SNA
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Mechanism for the Intermittency in the
Quasiperiodically Forced Ring Map

• In the RA of level k7, 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 Toda Oscillator
? Quasiperiodically Forced Toda Oscillator
Phase Diagram (?0.8, ?12)
Route a Intermittency Route b or c Interior
crises of SNA or CA Smooth Torus (Light Gray) T
and 2T CA (Black), SNA (Gray and Dark Gray)
? Appearance of Intermittent SNA
Smooth Torus
Intermittent SNA
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Mechanism for the Intermittency in the
Quasiperiodically Forced Toda Oscillator

• In the RA of level k7, 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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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-Shaped Unstable Set.