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

Q. Cao, M. Wiercigroch, E. E Pavlovskaia, J. M. T Thompson, C. Grebogi
2008 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
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 nondimensional measure of the span of the arch) changes the smooth load-deflection curve associated with snap-buckling into a discontinuous sawtooth. The smooth
more » ... wtooth. 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 aZ0. 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 aZ0.2 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.
doi:10.1098/rsta.2007.2115 pmid:17698466 fatcat:rjowlfme5jen7o7lfornl6k3xe