Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems

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

2. 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 2T CA (Black), SNA (Gray and Dark Gray)

3. _ ~ Phase Sensitivity Exponent to Characterize Strangeness of an Attractor  Phase Sensitivity with Respect to the Phase of Quasiperiodic Forcing: MeasuredbyCalculatingaDerivativex/alongaTrajectoryandFindingitsMaximumValue: (Taking the minimum value of N(x0,0) with respect to an ensemble of randomly chosen initial conditions) Phase Sensitivity Function: • 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

4. Typical Phase Diagrams in Quasiperiodically Forced Period-Doubling Systems Quasiperiodically Forced Hénon Map Quasiperiodically Forced Ring Map Tongues (near the Terminal Points of the Torus Doubling Bifurcation Lines) =0 and b=0.01 b=0.05 (a: Intermittency, b & c: Interior Crisis, d & e: Boundary Crisis)

5. _ _ _ ~ ~ ~ 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. *=0.584 726 781 Smooth Torus inside an AA for a=3.38 and =0.584 7 (x -0.059) Intermittent SNA filling the AA for a=3.38 and =0.584 75 (x -0.012,  19.5)

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

7. 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 Orbits The Intermittent Transition is Expected to Occur through Collision with an Unstable Orbit: •Smooth Unstable Torusx=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.

8. Metamorphoses of the Ring-Shaped Unstable Set • Thekth RA to a Smooth Torus e.g. k=6  RA: Composed of Stable Orbits with Period F6 (=8) inside the AA. • BirthofaRing-ShapedUnstableSet(RUS)viaaPhase-DependentSaddle-Node Bifurcation a=3.246, =0.446, k=6 • 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.) • Evolution of the Rings a=3.26, =0.46, k=6 • 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)

9. Change in the Shape and Size in the Rings a=3.326, =0.526, k=6 Each Ring:Composedof theLargeUnstable Part (Gray) and a Small Attracting Part (Black) • Quasiperiodic Limit a=3.326, =0.526, k=8 No. of Rings (=336): Significantly Increased Unstable Part (Gray): Dominant Attracting Part (Black): Negligibly Small Expectation: In the Quasiperiodic Limit, the RUS forms a Complicated Unstable Set Composed of Only Unstable Orbits

10. _ ~ Mechanism for the Intermittency • Smooth Torus (Black) and RUS (Gray) in the RA of Level k=8 (F8=21) 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) a=3.38, =0.5864, k=8 a=3.38, =0.5864, k=8

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

12. 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 Limit Solid Circles: RA of Level k=15

13. _ _ _ _ _ _ ~ ~ ~ ~ ~ ~ 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 • a=3.4443 • =0.55 • x -0.005  8.0 • a=3.4441 • =0.55 • x -0.018  1.4 Route c: CA Intermittent CA • a=3.43 • =0.48 • x 0.069 • a=3.44 • =0.48 • x 0.124

14. Boundary Crisis 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 • 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

15. AppearanceofHigher-OrderTongues Appearance of Similar Higher-Order Tongues Torus (Light Gray) SNA (Gray) CA (Black) (a*=3.569 945 ...) Band-Merging Transitions near the Higher-Order Tongues 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 • 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

16. 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 =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).

17. 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. a=3.4 =0.58 a=3.43 =0.58  2nd-Order Tongue In the Twice Iterated Map, when the Critical Curve L1 of Rank 1 Passes the Upper Basin Boundary, “Holes,” Leading to an Another Attractor, Appears inside the Basin. a=3.44 =0.14 a=3.45 =0.14

18. 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) 1.1Gradual Fractalization [T. Nishikawa and K. Kaneko, Phys. Rev. E 54, 6114 (1996)] 1.2Torus Collision [J.F. Heagy and S.M. Hammel, Physica D 70, 140 (1994)] 1.3Intermittent 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)] 1.4Blowout 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

19. _ _ _ _ ~ ~ ~ ~ Band-Merging Transition  Band-Merging Transition of Type I through Collision with the RUS 2T 1 SNA • a=3.43 • =0.165 • x -0.014  9.7 a=3.431 =0.16  Band-Merging Transition of Type II through Collision with the Unstable Parent Torus 2CA 1CA • a=3.32 • =0.202 • x 0.033 • a=3.32 • =0.198 • x 0.014