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Evidence for homoclinic orbits as a precursor to chaos in a magnetic pendulum. (English) Zbl 0607.70027
Experimental evidence is presented which supports the theory that homoclinic orbits in a Poincaré map associated with a phase space flow are precursors of chaotic motion. A permanent magnet rotor in crossed steady and time-varying magnetic fields is shown to satisfy a set of third order differential equations analogous to the forced pendulum or to a particle in a combined periodic and traveling wave force field. Critical values of magnetic torque and forcing frequency are measured for chaotic oscillations of the rotor and are found to be consistent with a lower bound for the existence of homoclinic orbits derived by the method of Melnikov. The fractal nature of the strange attractor is revealed by a Poincaré map triggered by the angular position of the rotor. Numerical simulations using the model also agree well with both theoretical and experimental criteria for chaos.

MSC:
70K50 Bifurcations and instability for nonlinear problems in mechanics
70K05 Phase plane analysis, limit cycles for nonlinear problems in mechanics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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