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Zones of chaotic behaviour in the parametrically excited pendulum. (English) Zbl 1232.70031

From the text: We consider the structural stability of three types of chaotic motion that occur in the parametric pendulum. Plotting the zones in parameter space reveals that tumbling chaotic motions, which fill much of the phase space, appear over a broad range of parameters. Within the zone of tumbling chaotic motions there are large areas in which this motion is the only stable motion and therefore this solution is relatively easy to find both numerically and experimentally. The fact that the region is comparatively large indicates that the response is robust with respect to small changes in the system parameters. On the other hand, oscillating and rotating chaotic motions appear only in narrow zones in the parameter space, coexisting with other motions, and are thus less structurally stable. It should perhaps be noted that for comparison with experimental results attention must be paid to the scaling, leading to the equation (1) \(\ddot \theta +\beta \dot \theta +(1+p\cos \omega t)\sin \theta=0\).

MSC:

70K50 Bifurcations and instability for nonlinear problems in mechanics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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