Koch, B. P.; Leven, R. W. Subharmonic and homoclinic bifurcations in a parametrically forced pendulum. (English) Zbl 0585.70022 Physica D 16, 1-13 (1985). 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. Cited in 32 Documents 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 Keywords:Poincaré map; chaos; parametrically forced pendulum; subharmonic and homoclinic bifurcations; Melnikov method; averaging methods; resonances; homoclinic orbits; unstable chaotic motions Citations:Zbl 0202.552 PDFBibTeX XMLCite \textit{B. P. Koch} and \textit{R. W. Leven}, Physica D 16, 1--13 (1985; Zbl 0585.70022) Full Text: DOI References: [1] Collet, P.; Eckmann, J. P., Iterated Maps on the Interval as Dynamical Systems (1980), Birkhäuser: Birkhäuser Basel · Zbl 0458.58002 [2] Misiurewicz, M., Ann. N.Y. Acad. Sci., 317, 348 (1980) [3] Smale, S., Bull. Amer. Math. Soc., 73, 747 (1967) [4] Moser, J., Stable and Random Motions in Dynamical Systems, (Ann. Math. Studies (1973), Princeton Univ. Press: Princeton Univ. 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