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Unformatted text preview: 82 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 . Many nonlinear systems also exhibit chaos , as does the forced pendulum for very large F . In a chaotic system, trajectories never settle into a limit cycle or other repeating pattern. Furthermore, the solutions are very sensitive to the initial conditions, so that if we start with two sets of conditions very close to each other (say 6 10 10 apart) at t = 0, within a relatively short time the solutions are a long way apart (several orders of magnitude greater, e.g., > 1). This makes the solution effectively unpredictable, even on a very accurate computer. 83...
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 Spring '13
 MRR
 Math, Equations, Critical Point, Equilibrium point, Stability theory, θ, Plane Autonomous Systems

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