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Subharmonic and homoclinic bifurcations in a parametrically forced pendulum. (English) Zbl 0585.70022

Summary: Depending on the parameters of a parametrically forced pendulum system the boundaries of subharmonic and homoclinic bifurcations are calculated on the basis of the Melnikov method and of averaging methods. It is shown that, as a parameter is varied, repeated resonances of successively higher periods occur culminating in homoclinic orbits. According to the theorem of S. Smale [Bull. Am. Math. Soc. 73, 747-792 (1967; Zbl 0202.552)] homoclinic bifurcation is the source of the unstable chaotic motions observed. For some selected parameter sets the theoretical predictions are tested by numerical calculations. Very good agreement is found between analytical and numerical results.

MSC:

70K40 Forced motions for nonlinear problems in mechanics
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
37G99 Local and nonlocal bifurcation theory for dynamical systems

Citations:

Zbl 0202.552
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References:

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