The elastic pendulum: A nonlinear paradigm

Abstract

A pendulum with an elastic instead of an inextensible suspension is the simplest realization of an autonomous, conservative, oscillatory system of several degrees of freedom with nonlinear coupling; it can also have an internal 1:2 resonance. A fairly complete study of this system at and near resonance is here undertaken by means of the ’’slow-fluctuation’’ approximation which consists in developing the x2y-type interaction into a trigonometric polynomial and keeping only the term with the slowest frequency. Extensive computations showed that up to moderately large amplitudes the approximate solutions were virtually as accurate as numerical integrations of the exact equations of motion. The slow-fluctuation equations of motion can be completely integrated by quadratures. Explicit solutions for amplitudes and phases are given in terms of elliptic functions, and can be linked to initial conditions. There exist two branches of purely periodic, harmonic, constant-amplitude motions which are orbitally stable but Liapunov unstable. The pure suspension motion is Liapunov unstable and remains orbitally stable only up to and including a critical amplitude; the standard ’’method of variational equations’’ leads to a slightly different stability criterion but is shown to be unreliable. In the dynamical neighborhood of the unstable pure suspension mode are motions which convert to it after infinite time. When a motion has an amplitude modulation minimum at or near zero, a phase reversal of the suspension takes place which is shown to be an artefact inherent in the description in terms of amplitudes and phases. In addition there is in the pendulum (but not in the exactly soluble system having the slow–fluctuation Hamiltonian) a fast phase transient which vitiates the slow-fluctuation technique for a few periods around the suspension amplitude minimum; this is the only restriction on the method. An appendix outlines formal isomorphisms between the elastic pendulum and the process of second-harmonic generation in nonlinear optics.

Nonlinear Normal Modes and Their Bifurcations for an Inertially Coupled Nonlinear Conservative System

“…Most of Pak's study is limited to using the harmonic balance method or direct numerical simulations along with Poincaré sections to understand the free response of the system. The 'spring pendulum', a system very similar to the spring-mass-pendulum system, has however received much attention in physics literature, serving as a paradigm for nonlinear dynamics and chaos [21][22][23], as well as a model for many nonlinear phenomena in complex systems [24]. For the same spring-pendulum model, Broer et al [25,26] have investigated both the 1:2 and the 1:1 resonance cases by using the equivariant singularity theory with distinguished parameters.…”

A generic, hard transition to chaos

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Sudden transition to chaos in plasma wave interactions

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