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Here we treat a pendulum called periodically forced oscillator. It is described by a differential equation written as
d2θ/dt2 = - γ dθ/dt - sin(θ) + a cos(ω t) where γ = 0.22, ω =1.0, and a = 2.7. Solving this equation numerically, a path θ(t) is derived. Sampling it every T (s), we can get two-dimensional discrete-time data (θ(nT), dθ(nT)/dt) (n=0,1,...). If we choose the period of external force (2π/ω) as sampling time T, we can get the same strange attractor as shown in "attractors of periodically forced pendulum" gallery. If we change the time when the sampling start, the structure of the attractor varies. The above animation is a collection of such sequences of attractors. The shape of attractor is changing with the same period as external force. You can observed the dynamics of the periodically forced pendulum in "periodically forced pendulum" simulator. |
The images used to make this animation are as follows:
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Periodically Forced Pendulum 2 Animation >>
Introduction to Chaos and Nonlinear Dynamics