θ_{o} = θ_{o} + Δ +
k sin θ_{o}
it follows -sin θ_{o} = Δ / k ≤ 1. Thus for k ≥ Δ there is attracting 0/1 orbit (it corresponds to the first intersection of the blue curve with diagonal). For larger k period doubling cascade takes place. Attracting 0/1 orbit loses its stability when derivative of the map 1 + k cos θ_{o} = -1 at hyperbola k^{2} = Δ^{2} + 4. The stable 1/1 orbit (it corresponds to the second intersection of the red curve with diagonal) is determined by θ_{o} + 2π = θ_{o} + Δ + k sin θ_{o}. |
Bifurcation (isoperiodic) diagram on the (Δ/2π, k/2π) plane.
Controls: Click mouse in window to find period p of the point. Click mouse + <Alt>(<Ctrl>) to Zoom In(Out) 2 times. 600x350 picture (1.5 times larger) with multithreading for multi-cores processors. |
Descending to k = 0 and θ/2π = p/q Arnold's tongues
correspond to resonant periodic orbits (p, q).
For k < 1 black regions correspond to quasi-periodic dynamics.
Measure of these regions decreases to zero for k → 1.
For k > 1 in black chaotic see there are windows of regular dynamics (Milnor's swallow) again. For k > Δ and k > 2π - Δ the map has two attracting points (see Fig.2 to the left). Depending on initial conditions iterations go to one or another attractor (bi-stability). |