1 Universality for the Intermittent Route to Strange Nonchaotic Attractors in Quasiperiodically Forced Systems W. Lim and S.-Y. Kim Kangwon National University.

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Presentation transcript:

1 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 TorusIntermittent SNA Property of SNAs: 1. No Sensitivity to Initial Condition (  <0) 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  .

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

4 Mechanism for the Intermittency 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  Quasiperiodically Forced Ring Map Phase Diagram (b=0.01) Smooth Torus Intermittent SNA 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

6 Mechanism for the Intermittency in the Quasiperiodically Forced Ring Map 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

7 Intermittent Route in the Quasiperiodically Forced Toda Oscillator  Quasiperiodically Forced Toda Oscillator Smooth Torus Intermittent SNA Phase Diagram (  =0.8,  1 =2) 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

8 Mechanism for the Intermittency in the Quasiperiodically Forced Toda Oscillator 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

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-Shaped Unstable Set.