1

Band-Merging Route to Strange Nonchaotic Attractors in Quasiperiodically Forced Systems

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

Quasiperiodically Forced 1D Map

),1(mod),1()2cos(

:1

1

nn

nnnn xxaxM .

215

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=0.053x=0.96

a=3.596=0.046x=0.159

Two-Band CA Single-Band CA

Investigation of BM Transitions in M2: Two-Band CA in M A Pair of Conjugate CA in M2

2

Route : Standard BM Transition of the CA through a Collision with the Smooth Unstable Parent TorusRoute : Standard BM Transition of the Strange Nonchaotic Attractor (SNA) through a Collision with the Smooth Unstable Parent TorusRoute : 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 RUSRoute a: Appearance of the Two-Band Intermittent SNARoute b: Attractor Widening Crisis of the SNA

State Diagrams near the Second Order Tongue

Magnified Phase Diagram

3

Basin Boundary Metamorphosis

In M2, the Smooth Doubled Torus with Two Bands Turns into a Pair of Conjugate Tori inside Their Absorbing Area Bounded by the Critical Curves Lk (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-F5 (=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 F5 (=5) (Unstable Part: Toward the Smooth Torus They may Interact.)

515.04.3

k

a

5146.0396.3

k

a

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 :

1 and 0,;/ 10111 FFFFFFF kkkkkk• Properties of the Quasiperiodically Forced Systems Obtained by Taking the Quasiperiodic Limit k .

Birth of a RUS Evolution of the Rings

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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.

161.0,43.3 a 162.0,43.3 a

716.7,067.0 x

Smooth Doubled Torus with Two Bands Single-Band SNA

479323161.0*

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Mechanism for the Band-Merging of the Smooth Torus

1597.0,43.3 a 15976.0,43.3 a

In the RA of level k=8, the Phase-Dependent Saddle-Node-Bifurcation between Smooth Torus and RUS on the Hole Boundary Occurs for (=0.159 750 121) when a=3.43. Appearance of F8 (=21) “Gaps”, where Single-Band Intermittent CAs Exist.

105.0x

*8

7

Band-Merging Route to SNA in Quasiperiodically Forced High-Dimensional Invertible Systems

Quasiperiodically Forced Hénon Map

),1(mod.

,2cos:

1

1

21

nn

nn

nnnn

bxyyxax

M

.2

15

12.0,17.1 a 127.0,17.1 a

60.4~029.0~1

Smooth Doubled Torus with Two Bands Single-Band SNA

718662126.0*

State Diagram for b=0.05

8

Quasiperiodically Forced Toda Oscillator

.2/)15()/(

,coscos1

12

21

ttaexx x

21.0,27 a 244.0,27 a

2.6~051.0~1

Smooth Doubled Torus with Two Bands Single-Band SNA

437953242.0*

State Diagram for =0.8 and 1=2

9

Quasiperiodically Forced Hodgkin-Huxley Oscillator

22 A/cm03.0,A/cm3.50 A

17.2~029.0~1

Smooth Doubled Torus with Two Bands Single-Band SNA

519451033.0*

State Diagram for Idc=100A/cm2 and f1=26Hz

.2/)15(/,2sin2sin

,,,;)(

)()(1)(

,

,

1221

43

fftftfAII

nhmxVxVxxVxV

dtdx

IEVgEVngEVhmg

IIIIIIdtdVC

dcext

xxx

extLLKKNaNa

extLKNaextion

22 A/cm0336.0,A/cm3.50 A

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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 aSmooth 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.