1 Band-Merging Route to Strange Nonchaotic Attractors in Quasiperiodically Forced Systems Woochang Lim and Sang-Yoon Kim Department of Physics Kangwon National University  Quasiperiodically Forced 1D Map  Band-Merging (BM) Transition of the Chaotic Attractor (CA) Through a Collision with the CA and a Smooth Unstable Parent Torus (Dashed Line), the “Standard” BM Transition of the CA Occurs. a=3.603  =0.053  x =0.96 a=3.596  =0.046  x =0.159 Two-Band CA Single-Band CA Investigation of BM Transitions in M 2 : Two-Band CA in M  A Pair of Conjugate CA in M 2

2 Route  : Standard BM Transition of the CA through a Collision with the Smooth Unstable Parent Torus Route  : Standard BM Transition of the Strange Nonchaotic Attractor (SNA) through a Collision with the Smooth Unstable Parent Torus Route  : Appearance of the Single Band SNA via a Collision with the Smooth Unstable Parent Torus (Heagy-Hammel Route) Route A: BM Transition of the Smooth Torus through a Collision with a Ring-Shaped Unstable Set (RUS) Route B(C): BM Transition of the SNA (CA) through a Collision with a RUS Route a: Appearance of the Two-Band Intermittent SNA Route b: Attractor Widening Crisis of the SNA State Diagrams near the Second Order Tongue Magnified Phase Diagram

3 Basin Boundary Metamorphosis In M 2, the Smooth Doubled Torus with Two Bands Turns into a Pair of Conjugate Tori inside Their Absorbing Area Bounded by the Critical Curves L k (k=1, …, 8). The Basins of Upper and Lower Tori are shown in Light Gray and Gray, Respectively. A Smooth Unstable Torus (Dashed Line) Lies on a Basin Boundary. Through a Breakup of the Absorbing Area via a Collision with the Smooth Unstable Parent Torus on the Basin Boundary, “Holes” of other basin of the counterpart Appear inside the Basins of the Smooth Attracting Tori. Through the Basin Boundary Metamorphosis, the Smooth Unstable Parent Torus Becomes Inaccessible from the Interior of Basin of the Upper and Lower Tori. a=3.46  =0.11 a=3.48  =0.13

4 Expectation: In the Quasiperiodic Limit, the RUS forms a Complicated Unstable Set Composed of Only Unstable Orbits Appearance of CA via Period-Doubling Bifurcations (PDBs) and Its Disappearance via a Boundary Crisis (Lower Gray Line: Period-F 5 (=5) Orbits Destabilized via PDBs) RUS of Level k=5: Composed of 5 Small Rings Each Ring: Composed of Stable (Black) and Unstable (Gray) Orbits with Period F 5 (=5) (Unstable Part: Toward the Smooth Torus  They may Interact.) Ring-Shaped Unstable Set  Rational Approximation (RA) Investigation of the BM 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  .  Birth of a RUS  Evolution of the Rings

5 Appearance of the SNA via a Band-Merging Transition Through a Collision with a Smooth Doubled Torus with Two Bands and Hole Boundary, BM Transition of the Smooth Torus Occurs, and then a Single-Band SNA Appears. Smooth Doubled Torus with Two Bands Single-Band SNA

6 Mechanism for the Band-Merging of the Smooth Torus In the RA of level k=8, the Phase-Dependent Saddle-Node-Bifurcation between Smooth Torus and RUS on the Hole Boundary Occurs for (= ) when a=3.43.  Appearance of F 8 (=21) “Gaps”, where Single-Band Intermittent CAs Exist.

7 Band-Merging Route to SNA in Quasiperiodically Forced High- Dimensional Invertible Systems  Quasiperiodically Forced Hénon Map Smooth Doubled Torus with Two Bands Single-Band SNA State Diagram for b=0.05

8  Quasiperiodically Forced Toda Oscillator Smooth Doubled Torus with Two Bands Single-Band SNA State Diagram for  =0.8 and  1 =2

9  Quasiperiodically Forced Hodgkin-Huxley Oscillator Smooth Doubled Torus with Two Bands Single-Band SNA State Diagram for I dc =100  A/cm 2 and f 1 =26Hz

10 Summary Investigation of the Band-Merging Route to SNA Using the Rational Approximation New Type of Band-Merging Transition for a Nonchaotic Attractor (Smooth Torus or SNA) as well as a Chaotic Attractor Occurs through the Collision with a Ring-Shaped Unstable Set. Particularly, a Single-Band SNA Appears via a New Band-Merging Transition of a Smooth Doubled Torus.  New Mechanism for the Birth of SNA Universal Band-Merging Route to SNA Band-Merging Route to SNA Found in the High-Dimensional Invertible Systems such as Quasiperiodically Forced Hénon Map, Toda Oscillator, and Neural System.