Quasiperiodicity & Mode Locking in The Circle Map Eui-Sun Lee Department of Physics Kangwon National University  Circle map  Lyapunov exponent  Winding number

Phase Diagram in The Circle Map 1/2 2/3 4/7 5/7 3/5 3/4 4/5 5/6 1/1 The regions the (Ω,K) where W occupies rational values, are called Arnold Tongues. has the 1D structure, and then ends at the critical point (K=1).

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.