title: Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics.
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, alpha, 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.
authors: Centre for Applied Dynamics Research, Department of Engineering, King's College, University of Aberdeen, Aberdeen AB24 3UE, UK.
authors: Qingjie
authors: Q
authors: Cao
authors: Marian
authors: M
authors: Wiercigroch
authors: Ekaterina E
authors: EE
authors: Pavlovskaia
authors: J Michael T
authors: JM
authors: Thompson
authors: Celso
authors: C
authors: Grebogi