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The parametrically excited pendulum: A paradigm of nonlinear systems

Clifford, Michael John; (1995) The parametrically excited pendulum: A paradigm of nonlinear systems. Doctoral thesis (Ph.D), UCL (University College London). Green open access

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Abstract

The parametrically excited pendulum is a simple nonlinear system that exhibits a plethora of nonlinear phenomena. The steady state solutions can be subdivided into hanging, inverted, rotating, and non-rotating solutions. These in turn undergo bifurcations such as symmetry breaking, saddle-node, period-doubling, sub-critical, super-critical, and catastrophic bifurcations. These bifurcations are studied in detail by a variety of analytical and numerical techniques including perturbation methods, harmonic balance, method of strained parameters, braid and knot theory, Melnikov energy methods, Runge-Kutta numerical integration, Poincare sections, path following, bifurcation following, and cell mapping. For non-rotating solutions, the parametrically excited pendulum may be considered as a system which permits escape from a symmetric potential well under parametric excitation. This simple statement allows a large body of existing theory on escape systems to be applied directly to the pendulum. Parameter zones where no major non-rotating orbits exist are termed "escape zones", and are predicted by applying various analytical techniques. The results are compared with the numerically determined bifurcation diagrams, and are assessed in terms of engineering integrity and the fractal nature of basin boundaries. The implications for real physical systems are considered in terms of safe operating parameters. Braid and knot theory is applied to a particular horseshoe which is conjectured to exist in the invariant manifolds of the hilltop saddles for the system. This analysis provides considerable insight into subharmonic bifurcational activity, and the role this activity plays in the overall bifurcational structure. It is argued that this combined analytical and numerical approach gives a more complete picture of the dynamics of the parametrically excited pendulum than either method would alone.

Type: Thesis (Doctoral)
Qualification: Ph.D
Title: The parametrically excited pendulum: A paradigm of nonlinear systems
Open access status: An open access version is available from UCL Discovery
Language: English
Additional information: Thesis digitised by ProQuest.
Keywords: Pure sciences
URI: https://discovery.ucl.ac.uk/id/eprint/10105742
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