Dynamic interactions of traveling waves propagating in a linear chain with an local essentially nonlinear attachment

Abstract

We study strongly nonlinear dynamical interactions between traveling waves propagating in a linear spring-mass chain with a strongly nonlinear, lightweight local attachment. We analyze the dynamics of this system by constructing a reduced model in the form of a strongly nonlinear integro-differential equation with inhomogeneous terms representing local and non-local interactions between the chain and the nonlinear attachment. Then we construct homoclinic and subharmonic Melnikov functions and prove the existence of chaotic motions and subharmonic periodic orbits in the combined chain-attachment system. In the limit of weak coupling between the particles of the chain we study the bifurcations that generate stable-unstable pairs of subharmonic motions. Generalizations of the methodology to a more general class of linear systems with local essentially nonlinear attachments are discussed. This work provides further evidence that the break of symmetry of an otherwise linear chain by a strongly nonlinear (even lightweight) attachment can give rise to complex (even chaotic) dynamics. The underlying dynamical mechanism of this complexity is nonlinear resonant energy transfer from the traveling waves to the nonlinear attachment. The presented results contribute towards the study of the dynamic and resonance interactions of waves propagating in extended media with strongly nonlinear local attachments.