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Mechanism for Boundary Crises in Quasiperiodically Forced Systems

Woochang Lim and Sang-Yoon Kim Department of PhysicsKangwon National University

Quasiperiodically Forced 1D Map

Route A: Standard Boundary Crises (BC) of the Chaotic Attractor (CA) through a Collision with the Smooth Unstable TorusRoute B: Standard BC of the Strange Nonchaotic Attractor (SNA) through a Collision with the Smooth Unstable TorusRoute C: BC of the Smooth Torus via a Collision with the Ring-Shaped Unstable Set (RUS)Route D (E): BC of the SNA (CA) via a Collsion with the RUSSmooth Torus (Light Gray): T and 2TCA (Black), SNA (Gray and Dark Gray)

Phase Diagram

),1(mod

,2cos:

1

21

nn

nnn xaxM .

2

15

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Standard Boundary Crises of a CA

Through a Collision with the CA and a Smooth Unstable Torus (Dashed Line) on the Basin Boundary, the BC of the CA Occurs.

This BC Corresponds to a Natural Generalization of the BC Occurring for the Unforced Case (=0). “Standard” BC

a=1.19, =0.315 a=1.265, =0.3525

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Basin Boundary Metamorphosis

As a Result of the Breakup of the Absorbing Area via a Collision with the Smooth Unstable Torus on the Basin Boundary, “Holes” Appear inside the Basin of the Smooth Attracting Torus.

Through the Basin Boundary Metamorphosis, the Smooth Unstable Torus Becomes Inaccessible from the Interior of Basin of the Smooth Torus.

a=1.05, =0.355 a=1.187, =0.4235

Rational Approximation (RA)

• Investigation of the BC 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

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Ring-Shaped Unstable Set

7,3245.0,989.0 ka

7,3265.0,993.0 ka

Birth of a 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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Boundary Crises via Collisions with Holes

When Following the Route C, D, and E in the Phase Diagram, Boundary Crises of Smooth Torus, SNA, and CA Occur through Collision with HolesInside the Basin.

Smooth Torus SNA CA

405.0,227.1 a43.0,2327.1 a445.0,18.1 a

077.1,038.0 x

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Mechanism for the Boundary Crisis of the Smooth Torus

427.0,18.1 a 43.0,18.1 a 4309.0,18.1 a

In the RA of level k=7, when passing a threshold value of , RUS lies on the hole boundary.Eventually, the phase-dependent SNB between smooth torus and RUS on the hole boundary occurs for * (=0.430 854 479). Appearance of gaps, where the former attractor no longer exists.

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Mechanism for the Boundary Crisis of SNA

43.0,207.1 a 43.0,21.1 a

In the RA of level k=7, the BC of the Chaotic Component of the RA of the SNA and RUS on the hole boundary occurs for a* (=1.208 945 689). Appearance of gaps, where the former attractor no longer exists.

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Summary

• Investigation of the Boundary Crises Using the Rational Approximation

New Type of Boundary Crisis Occurs through the Collision with a Ring-ShapedUnstable Set.

As a Result, a Nonchaotic Attractor (Smooth Torus or SNA) as well as a chaoticAttractor is Abruptly Destroyed.