A reversible bifurcation analysis of the inverted pendulum Broer, H.W.; Hoveijn, I.; Noort, M. van Citation for published version (APA): Broer, H. W., Hoveijn, I., & Noort, M. V. (1998). A reversible bifurcation analysis of the inverted pendulum. AbstractThe inverted pendulum with a periodic parametric forcing is considered as a bifurcation problem in the reversible setting. Parameters are given by the size of the forcing and the frequency ratio. Normal form theory provides an integrable approximation of the Poincar6 map generated by a planar vector field. Genericity of the model is studied by a perturbation analysis, where the spatial symmetry is optional. Here equivariant singularity theory is used.

The problem of this article is the characterization of equivalence classes and their versal deformations for reversible and reversible Hamiltonian matrices. In both cases the admissible transformations form a subgroup G of Gl(m). Therefore the Gl(m)-orbits of a given matrix may split into several G-orbits. These orbits are characterized by signs. For each sign we have a normal form and a corresponding versal deformation. The main tool in the characterization is reduction to the semi simple case.1996 Academic Press, Inc.

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This paper is concerned with the global coherent (i.e., non-chaotic) dynamics of the parametrically forced pendulum. The system is studied in a 1 1 2 degree of freedom Hamiltonian setting with two parameters, where a spatio-temporal symmetry is taken into account. Our explorations are restricted to large regions of coherent dynamics in phase space and parameter plane. At any given parameter point we restrict to a bounded subset of phase space, using KAM theory to exclude an infinitely large region with rather trivial dynamics. In the absence of forcing the system is integrable. Analytical and numerical methods are used to study the dynamics in a parameter region away from integrability, where the analytic results of a perturbation analysis of the nearly integrable case are used as a starting point. We organize the dynamics by dividing the parameter plane in fundamental domains, guided by the linearized system at the upper and lower equilibria. Away from integrability some features of the nearly integrable coherent dynamics persist, while new bifurcations arise. On the other hand, the chaotic region increases.