Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics

Qingjie Cao, Marian Wiercigroch, Ekaterina E. Pavlovskaia, John Michael Tutill Thompson, Celso Grebogi

Research output: Contribution to journalArticle

92 Citations (Scopus)

Abstract

In a recent paper we examined a model of an arch bridge with viscous damping subjected to a sinusoidally varying central load. We showed how this yields a useful archetypal oscillator which can be used to study the transition from smooth to discontinuous dynamics as a parameter, a, tends to zero. Decreasing this smoothness parameter (a non-dimensional measure of the span of the arch) changes the smooth load deflection curve associated with snap-buckling into a discontinuous sawtooth. The smooth snap-buckling curve is not amenable to closed-form theoretical analysis, so we here introduce a piecewise linearization that correctly. fits the sawtooth in the limit at alpha=0. Using a Hamiltonian formulation of this linearization, we derive an analytical expression for the unperturbed homoclinic orbit, and make a Melnikov analysis to detect the homoclinic tangling under the perturbation of damping and driving. Finally, a semi-analytical method is used to examine the full nonlinear dynamics of the perturbed piecewise linear system. A chaotic attractor located at alpha=0.2 compares extremely well with that exhibited by the original arch model: the topological structures are the same, and Lyapunov exponents (and dimensions) are in good agreement.

Original languageEnglish
Pages (from-to)635-652
Number of pages18
JournalPhilosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences
Volume366
Issue number1865
Early online date13 Aug 2007
DOIs
Publication statusPublished - 28 Feb 2008

Keywords

  • Melnikov method
  • piecewise linearization
  • saddle-like singularity
  • homoclinic-like orbit

Cite this