Ml 2 θ θ mgl θ sin θ mkl θ 2 from 113 81 so the

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ml 2 ˙ θ ¨ θ + mgl ˙ θ sin θ = - mkl ˙ θ 2 (from (11.3)) 0 , 81
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so the energy is decreasing. (Without the damping term we would have had ˙ E = 0.) This allows us to deduce the portrait above from the undamped portrait of § 11.3, because the solution continuously moves to curves of lower energy. A more interesting example is the van der Pol oscillator ¨ x = - x - ( x 2 - 1) ˙ x. For large x we have damping (because x 2 - 1 > 0); but for small x we have “negative damping” ( x 2 - 1 < 0). The only equilibrium is at the origin and is an unstable focus. The phase portrait looks like this: So all trajectories (whether starting from small or large x , except for the one at the origin itself) tend towards a limit cycle . Therefore, after an initial transient the system always settles down into this finite amplitude oscillation. If we had studied only the linear stability near the origin we would have concluded that disturbances grow exponentially; the fact that this growth is in fact limited is useful in practical situations. 11.5 The Forced Pendulum Finally, add time-periodic forcing to the pendulum: ¨ θ = - g l sin θ - k ˙ θ + F cos Ω t where Ω is the forcing frequency and F its amplitude. This is no longer an autonomous system (unlike in § 11.1), because t appears in the governing equation in addition to θ and ˙ θ . The phase “plane” becomes three-dimensional and consequently much more elaborate: for instance, trajectories can now twist around each other, which is prevented in two dimensions by the rule that trajectories cannot cross except at equilibrium points. When F is small, we obtain a resonant response as described in § 2.2. But for larger forcing amplitudes the system exhibits much more complicated behaviour because the nonlinearity of the sin θ term interacts with the forcing. Eventually chaos results. 82
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For small values of F , a graph of oscillation amplitude | A | versus forcing frequency Ω has a simple resonant peak. As F is increased, the oscillations of the pendulum become larger; the restoring force - ( g/l ) sin θ is smaller than the linear approximation - ( g/l ) θ that applies for small F , so the period of the oscillations increases. This means that the forcing frequency Ω required to produce resonance decreases for larger | A | , so the peak bends backwards. When F becomes larger still, the oscillation amplitude can take more than one value for a range of Ω. Suppose that we slowly increase Ω from small values: then the amplitude will move along the lower curve shown until it has to “jump” to the upper curve. But when Ω is slowly decreased again the amplitude moves back in a different way, sticking to the upper curve until it has to “fall” down to the lower one. This phenomenon, whereby the solution of a system depends not just on its parameters but also on the history of those parameters, is known as hysteresis .
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