Attractors of Periodically Forced Pendulum

An attractor of periodically forced pendulum. The explanation is obtained in "periodically forced pendulum" animation.
The parameters are γ = 0.22 , ω =1.0, and a = 2.7.

The next figure is a three band attractor of the same dynamical system.
The parameters are γ=1/√15, ω=0.65, and a=0.728.

When the above three band attractor merges, one band attractor appears.
The parameters are γ=1/√15, ω=0.65, and a=0.75.
The bifurcation takes place at about a=0.728384.

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Introduction to Chaos and Nonlinear Dynamics