Sensitivity to Initial Conditions in Chaos

While there is no agreement on a definition of chaos, sensitivity to initial conditions is usually considered an important marker. This Demonstration compares plots of a driven pendulum in a chaotic regime, started at two slightly different positions, to the same pendulum restricted to a regime of periodic orbits. The periodic regime varies the periodic driving force amplitude (Subscript[f, 0])  from 0.05 to 0.5 instead of from 0.1 to 1.0. The larger range includes a number of periodic orbits as well as chaotic ones.

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DETAILS

Snapshot 1: quick coalescence to a periodic attractor

Snapshot 2: rapid divergence of trajectories in the chaotic regime

Snapshot 3: periodic attractor at a later time

Snapshot 4: trajectories can converge and diverge again; this is a period-two trajectory

The equation of motion is θ''+γ θ'+sin(θ)=(Subscript[f, 0])cos((Subscript[ω , d])t), with γ =0.2 and (Subscript[ω, d])=0.6.

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Contributed by: Bruce Hawkins

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