1. 1 Strange Nonchaotic Attractors in Quasiperiodically Forced Period-Doubling Systems  Quasiperiodically Forced Systems  : 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 TorusSNA (Intermediate State) Chaotic Attractor Sang-Yoon Kim (KWNU, UMD)

2. 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. 3 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: 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 (Taking the minimum value of  N (x 0,  0 ) with respect to an ensemble of randomly chosen initial conditions) ~ _

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

5. 5 Intermittent Route to SNAs  Absorbing Area (AA) in the Quasiperiodically Forced Logistic Map M M: Noninvertible [ detDM=0 along the Critical Curve L 0 ={x=0.5}] Images of the Critical Curve x=0.5 [i.e., L k =M k (L 0 ): 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 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)  * =0.584 726 781 ~ _ ~ _ ~ _

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

7. 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 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.

8. 8 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 F 6 (=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 F 6 (=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-F 6 (=8) Orbits Destabilized via PDBs)

9. 9 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 Increased Unstable Part (Gray): Dominant Attracting 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

10. 10 Mechanism for the Intermittency  Smooth Torus (Black) and RUS (Gray) in the RA of Level k=8 (F 8 =21) a=3.38,  =0.5864, 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 -0.09) ~ _

11. 11 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 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. 12 Transition from SNA to CA  Average Lyapunov Exponents (in the x-direction) in the RAs =  p +  c ;  p(c) : “Weighted” Lyapunov Exponents of the Periodic (Chaotic) Component  p(c) = p(c) M p(c) ; M p(c) : Lebesgue Measure in  and 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. 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 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 ~ _ ~ _ ~ _

14. 14  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

15. 15 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...)

16. 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. 17 Basin Boundary Metamorphoses  Main Tongue When the Critical Curve L 1 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 L 1 of Rank 1 Passes the Upper Basin 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

18. 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) 1.1 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

19. 19 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