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

2. 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).

3. 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.

4. 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 = ±

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

6. 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.