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1 Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems Quasiperiodically Forced Systems 1) (mod ), , ( 1 1 n n n n n x F x : Irrational No. Typical Appearance of Strange Nonchaotic Attractors (S NAs) Property of SNAs: 1. No Sensitivity to Initial Condition (<0) 2. Fractal Phas e Space Structure Smooth Torus SNA (Intermediate State) Chaotic Attractor Sang-Yoon Kim (KWNU, UMD)

Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems

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Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems. Sang-Yoon Kim (KWNU, UMD).  Quasiperiodically Forced Systems. : Irrational No.  Typical Appearance of Strange Nonchaotic Attractors (SNAs). Smooth Torus. SNA (Intermediate State). - PowerPoint PPT Presentation

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Page 1: Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems

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Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems

Quasiperiodically Forced Systems

1). (mod ),,( 11 nnnnn xFx

: Irrational No.

Typical Appearance of Strange Nonchaotic Attractors (SNAs)

Property of SNAs: 1. No Sensitivity to Initial Condition (<0) 2. Fractal Phase Space Structure

Smooth Torus SNA(Intermediate State)

Chaotic Attractor

Sang-Yoon Kim (KWNU, UMD)

Page 2: Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems

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Typical Dynamical Transitions in Quasiperiodically Forced Period-Doubling Systems

Quasiperiodically Forced Logistic Map

Phase Diagram

Main Interesting Feature

“Tongue,” where Rich Dynamical Transitions Occur: • Route a Intermittency • Route b or c Interior Crisis of SNA or CA • Route d or e Boundary Crisis of Torus or SNA

(All These Dynamical Transitions May Occur through Collision with a New Type of “Ring-Shaped Unstable Set.”)

Smooth Torus (Light Gray): T and 2TCA (Black), SNA (Gray and Dark Gray)

1), (mod

),1()2cos(:

1

1

nn

nnnn xxaxM .

2

15

Page 3: Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems

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Phase Sensitivity Exponent to Characterize Strangeness of an Attractor

Phase Sensitivity with Respect to the Phase of Quasiperiodic Forcing:

Measured by Calculating a Derivative x/ along a Trajectory and Finding its Maximum Value:

n

NnN

x0max

Phase Sensitivity Function: Nx

N ),( 00

min

• Smooth Torus (a=3.38, =0.584 7) N: Bounded No Phase Sensitivity

• SNA (a=3.38, =0.584 75) N ~ N: Unbounded [( 19.5): PSE] Phase Sensitivity Strange Geometry

(Taking the minimum value of N(x0,0) with respect to an ensemble of randomly chosen initial conditions)

~_

Page 4: Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems

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Typical Phase Diagrams in Quasiperiodically Forced Period-Doubling Systems

Quasiperiodically Forced Hénon Map

1). (mod

,

),2cos(

1

1

21

nn

nn

nnnn

bxy

yxax

Quasiperiodically Forced Ring Map

1). (mod

),2sin()2/(

,1) (mod )2cos(

)sin()2/(

1

1

1

nn

nnn

n

nnnn

xabyy

byxaxx

=0 and b=0.01b=0.05

(a: Intermittency, b & c: Interior Crisis, d & e: Boundary Crisis)

Tongues (near the Terminal Points of the Torus Doubling Bifurcation Lines)

Page 5: Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems

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Intermittent Route to SNAs

Absorbing Area (AA) in the Quasiperiodically Forced Logistic Map M

M: Noninvertible [ detDM=0 along the Critical Curve L0={x=0.5}]

Images of the Critical Curve x=0.5 [i.e., Lk=Mk(L0): Critical Curve of Rank k]:Used to Define a Bounded Trapping Region inside the Basin of Attraction.

The AA determines the Global Structure of a Newly-Born Intermittent SNA.

Smooth Torus inside an AAfor a=3.38 and =0.584 7(x -0.059)

Intermittent SNA filling the AAfor a=3.38 and =0.584 75(x -0.012, 19.5)

*=0.584 726 781

~_ ~_ ~_

Page 6: Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems

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Global Structure of an Intermittent SNAThe Global Structure of the SNA may be Determined by the Critical Curves Lk.

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Rational Approximations

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 .

Unstable OrbitsThe Intermittent Transition is Expected to Occur through Collision with an Unstable Orbit:

• Smooth Unstable Torus x=0 (developed from the unstable fixed point for the unforced case): Outside the AA No Interaction with the Smooth Attracting Torus

• Ring-Shaped Unstable Set (without correspondence for the unforced case) Using the RA, a New Type of Ring-Shaped Unstable Set that Interacts with the Smooth Torus is found inside the AA.

1 and 0,;/ 10111 FFFFFFF kkkkkk

2/)15(

Page 8: Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems

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Metamorphoses of the Ring-Shaped Unstable Set The kth RA to a Smooth Torus e.g. k=6 RA: Composed of Stable Orbits with Period F6 (=8) inside the AA.

a=3.246, =0.446, k=6

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

• RUS of Level k=6: Composed of 8 Small Rings

Each Ring: Composed of Stable (Black) and Unstable (Gray) Orbits with Period F6 (=8)

(Unstable Part: Toward the Smooth Torus They may Interact.)

a=3.26, =0.46, k=6

Evolution of the Rings

• Appearance of Chaotic Attractor (CA) via Period-Doubling Bifurcations (PDBs) and Its Disappearance via a Boundary Crisis

(Upper Gray Line: Period-F6 (=8) Orbits Destabilized via PDBs)

Page 9: Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems

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a=3.326, =0.526, k=6

a=3.326, =0.526, k=8

Change in the Shape and Size in the Rings

Quasiperiodic Limit

No. of Rings (=336): Significantly IncreasedUnstable Part (Gray): DominantAttracting Part (Black): Negligibly Small

Each Ring: Composed of the Large Unstable Part (Gray) and a Small Attracting Part (Black)

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 Smooth Torus (Black) and RUS (Gray) in the RA of Level k=8 (F8=21)

a=3.38, =0.5864, k=8 a=3.38, =0.5864, k=8

a=3.38, =0.586, k=8 a=3.38, =0.586, k=8

Phase-Dependent Saddle-Node Bifurcation for 8=0.586 366 Appearance of Gaps, Filled by Intermittent Chaotic Attractors RA to the Whole Attractor: Composed of Periodic and Chaotic (in 21 Gaps) Components Average Lyapunov Exponent < 0 (<x> -0.09)~_

Page 11: Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems

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• Algebraic Convergence of the Phase-Dependent SNB Values k (up to k=15) to its Limit Value *(=0.584 726 781) of Level k.

Quasiperiodic Limit

* ,|~| kkkk F

• A Dense Set of Gaps (Filled with Intermittent CAs)

Using This RA, One Can Explain the Intermittent Route to SNA.

a=3.38, =0.584 75

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Transition from SNA to CA

Average Lyapunov Exponents (in the x-direction) in the RAs

<x>= p + c;

p(c): “Weighted” Lyapunov Exponents of the Periodic (Chaotic) Component

p(c) = <x>p(c) Mp(c);

Mp(c): Lebesgue Measure in and <x>p(c): Average Lyapunov Exponent of the Periodic (Chaotic) Component

Chaotic Transition (c=0.5848)

Solid Line: Quasiperiodic LimitSolid Circles: RA of Level k=15

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Other Dynamical Transitions in the Tongues Interior Crisis

Attractor-Widening Interior Crisis Occurs through Collision with the RUS:

Route b: SNA (Born via Gradual Fractalization) Intermittent SNA

Route c: CA Intermittent CA

a=3.4441=0.55

x -0.018 1.4

~_

a=3.4443=0.55

x -0.005 8.0

~_

a=3.44=0.48

x 0.124~_

a=3.43=0.48

x 0.069~_

~_~_

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Boundary Crisis

Boundary Crisis of Type I (Heavy Solid Line) through Collision with the RUS • Route d: Smooth Torus Divergence • Route e: SNA (Born via Gradual Fractalization) Divergence

Boundary Crisis of Type II (Heavy Dashed Line) through Collision with the Smooth Unstable Torus • Route (): CA (SNA) Divergence

Double Crises near the Vertex Points

T: Torus, S: SNA, C: CA, D: Divergence,Dashed Line: Birth of SNA via Gradual Fractalization, Solid Line: Transition to Chaos, Dash-Dotted Line: Basin Boundary Metamorphosis Line, Dotted Line: Interior Crisis Line

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Appearance of Higher-Order Tongues Appearance of Similar Higher-Order Tongues

Band-Merging Transitions near the Higher-Order Tongues

• Hard Band-Merging Transition (Heavy Solid Line) via Collision with the RUS• Soft Band-Merging Transition (Heavy Dashed Line) via Collision with an Unstable Parent Torus• Double Crises near the Vertex Points

2T: Doubled Torus, S & 2S: SNA, C & 2C: CA,Dashed Line: Birth of SNA via Gradual Fractalization, Solid Line: Transition to Chaos, Dash-Dotted Line: Basin Boundary Metamorphosis Line, Dotted Line: Interior Crisis Line

Torus (Light Gray)SNA (Gray)CA (Black)

(a*=3.569 945 ...)

Page 16: Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems

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Summary Using the RA , the Quasiperiodically Forced Logistic Map has been Investigated: Appearance of a New Type of Ring-Shaped Unstable Sets via Phase-Dependent Saddle-Node Bifurcations near the Tongues Occurrence of Rich Dynamical Transitions such as the Intermittency, Interior and Boundary Crises, and Band-Merging Transitions through Interaction with the RUS Such Dynamical Transitions: “Universal,” in the Sense that They Occur Typically in a Large Class of Quasiperiodically Forced Period-Doubling Systems

• Phase Diagram of the Quasiperiodically Forced Toda Oscillator.2/)15()/( ,coscos1 1221 ttaexx x

=0.8 and 1=2

S.-Y. Kim, W. Lim, and E. Ott, “Mechanism for the Intermittent Route to Strange Nonchaotic Attractors,” nlin.CD/0208028 (2002).

Page 17: Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems

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Basin Boundary Metamorphoses Main Tongue

When the Critical Curve L1 of Rank 1 Passes the Upper Basin Boundary (x=1),“Holes,” Leading to Divergent Trajectories, Appears inside the Basin.

2nd-Order Tongue

In the Twice Iterated Map, when the Critical Curve L1 of Rank 1 Passes the UpperBasin Boundary, “Holes,” Leading to an Another Attractor, Appears inside the Basin.

a=3.4 =0.58

a=3.43 =0.58

a=3.44 =0.14

a=3.45 =0.14

Page 18: Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems

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Dynamical Transitions in Quasiperiodically Forced Systems

Rich dynamical transitions in the quasiperiodically forced systems have been reported:

1. Transitions from a Smooth Torus to a Strange Nonchaotic Attractor (SNA) 1.1 Gradual Fractalization [T. Nishikawa and K. Kaneko, Phys. Rev. E 54, 6114 (1996)]

1.2 Torus Collision [J.F. Heagy and S.M. Hammel, Physica D 70, 140 (1994)]

1.3 Intermittent Transition [A. Prasad, V. Mehra, and R. Ramaswamy, Phys. Rev. Lett. 79, 4127 (1997), A. Witt, U. Feudel, and A. Pikovsky, Physica D 109, 180 (1997)]

1.4 Blowout Transition [C. Grebogi, E. Ott, S. Pelikan, and J. A. Yorke, Physica D 13, 261 (1984), T. Yalcinkaya and Y.-C. Lai, Phys. Rev. Lett. 77, 5039 (1996)]

2. Crises for the SNA and Chaotic Attractor (CA) 2.1 Band-Merging Crisis [O. Sosnovtseva, U. Feudel, J. Kurths, and A. Pikovsky, Phys. Lett. A 218, 255 (1996)]

2.2 Interior Crisis [A. Witt, U. Feudel, and A. Pikovsky, Physica D 109, 180 (1997)]

2.3 Boundary Crisis [H.M. Osinga and U. Feudel, Physica D 141, 54 (2000)]

However, the Mechanisms for such Transitions are Much Less Understood than those in the Unforced or Periodically Forced systems. Illumination of the Mechanisms for the Dynamical Transitions: Necessary

Page 19: Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems

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Band-Merging Transition Band-Merging Transition of Type I through Collision with the RUS

2T 1 SNA

2CA 1CA

a=3.431=0.16

a=3.43=0.165

x -0.014 9.7

~_

a=3.32=0.202

x 0.033~_

a=3.32=0.198

x 0.014~_

~_

Band-Merging Transition of Type II through Collision with the Unstable Parent Torus