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Quasiperiodicity & Mode Locking in The Circle Map. Eui-Sun Lee Department of Physics Kangwon National University. Circle map. Lyapunov exponent. Winding number. has the 1D structure, and then ends at the critical point (K=1). Phase Diagram in The Circle Map. 2/3. 1/2.
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Quasiperiodicity & Mode Locking in The Circle Map Eui-Sun Lee Department of Physics Kangwon National University Circle map Lyapunov exponent Winding number
has the 1D structure, and then ends at the critical point (K=1). Phase Diagram in The Circle Map 2/3 1/2 4/7 5/7 4/5 1/1 3/4 5/6 3/5 • The regions the (Ω,K) where W occupies rational values, are called Arnold Tongues.
Devil’s Staircase In the subcritical region(0<K<1), the plot of W vs. Ω show the devils staircase. The winding No. is locked in at every single rational No. in a nonzero interval of Ω, and the plateaus exist densely.
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 .
Bifurcation Structure in the Arnold Tongues Swallow tail structure in the Arnold tongue exhibit self-similarity, and period-doubling transition to chaos occurs
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.
