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Quasiperiodicity & Mode Locking in The Circle Map

Lyapunov exponent

Circle map

Eui-Sun Lee

Department of Physics

Kangwon National University

.1mod,)θ π 2 sin(π2

θ)θ(θ nnn1n

Kf

.|)θ('|ln1

limσ1

0j

t

jt

ft

Winding number

.n

θθlimK),W( 0n

n

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Phase Diagram in The Circle Map

Gray No. irrational:W0,σ:statedicQuasiperioWhite 0σ:stateChaotic

Black No. rational : W0, σ :state Periodic

• The regions the (Ω,K) where W occupies rational values, are called Arnold Tongues.

2/)15(W :No. ngmean windi-golden having state dicquasiperio of )Wpath(The

1/2 2/3

1/13/43/5 4/5 5/65/7

W

4/7

• has the 1D structure, and then ends at the critical point (K=1).W

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Devil’s Staircase In the subcritical region(0<K<1), the plot of W vs. Ω show the devils staircase.

3

1

4

12

1

1

0

1

0

4

1 3

12

1

1

0

The winding No. is locked in at every single rational No. in a nonzero interval of Ω,and the plateaus exist densely.

2

1

3

1

4

1

1

0

5

1

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Measure of Mode locking

As K Increases toward 1 from 0 in The Subcritical Region, The Measure(M) of The Mode-locked State Increases Monotonically.

Near ΔK(=1-K) decreases, 1-M exhibits the power law scaling, 1-M~ΔK β

,where β = 0.305 ± 0.004 .

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Bifurcation Structure in the Arnold Tongues

Swallow tail structure in the Arnold tongue exhibit self-similarity, and period-doubling transition to chaos occurs

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Summary

1. The region in the (Ω,K) space where winding No. of the periodic state is locked in a single rational No., is called Arnold Tongues.

2. The inverse of golden-mean quasiperiodic state path has the 1D structure, and then ends at the critical point (K=1).

3. As K increases toward1 from 0 in the subcritical region, the measure(M) of the mode-locked state increases monotonically.

4. Swallow tail structure in the Arnold tongue exhibit self-similarity, and period-doubling transition to chaos occurs.