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Jump phenomena resulting in unpredictable dynamics in the driven damped pendulum

Authors
Journal
International Journal of Non-Linear Mechanics
0020-7462
Publisher
Elsevier
Publication Date
Volume
31
Issue
2
Identifiers
DOI: 10.1016/0020-7462(95)00050-x
Keywords
  • Pendulum Motion
  • Jump Phenomena
  • Bifurcations
  • Fractal Basin

Abstract

Abstract When an external force is applied to the pendulum, it leaves the equilibrium state, and after some transient, starts to oscillate in a steady-state manner. As parameters are varied, the system can undergo a series of bifurcations. One of the simplest forms of instability is the saddle-node bifurcation, which results in a jump phenomenon. This is often a purely deterministic event in which the pendulum jumps to resonance in a predictable manner. We show, however, that if the steady-state solution finds itself on a fractal basin boundary at the point of instability, we cannot predict the dynamics of the jump. We present resonance response diagrams, as the frequency is varied, to illustrate this behaviour.

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