Localization in finite vibroimpact chains: Discrete breathers and multibreathers

Abstract

We explore dynamics of discrete breathers and multi-breathers in finite one-dimensional chain. The model involves parabolic on-site potential with rigid constraints and linear nearest-neighbor coupling. The rigid non-ideal impact constraints are the only source of nonlinearity and damping in the model. The model allows derivation of exact analytic solutions for the breathers and multibreathers with arbitrary set of localization sites, both in conservative and forced-damped settings. We choose periodic boundary conditions; exact solutions for other types of the boundary conditions are also possible. Local character of the nonlinearity allows explicit derivation of a monodromy matrix for the breather solutions. Consequently, a stability of the derived breather and multibreather solutions can be efficiently studied in the framework of simple methods of linear algebra, and with rather moderate computational efforts. We demonstrate that finitness of the chain fragment and proximity of the localization sites strongly effect existence and stability patterns of these localized solutions. PACS numbers 05. 45.Yv, 63.20.Pw, 63.20.Ry

Periodic Motions of Coupled Impact Oscillators

“…Nondissipative impacts occur for y n (t) = −1 and give rise to impulsive reaction forces λ n (t). This configuration differs from the case of a symmetric local vibroimpact potential considered in [19,31] which introduces an additional barrier above the chain. The chain is allowed to oscillate above a straight obstacle.…”

“…For nonsmooth modes having certain patterns, simple nontrivial solutions of (12)- (19) can be found in the form (20) if some multiple of π/T belongs to the phonon band.…”

“…This has been achieved in a number of works in the case of breathers [17,18,37] and for nonsmooth modes close to grazing linear normal modes [27]. In the work [19], the analysis of [31] has been extended to several impacting particles, but the verification of the inequality constraints is still an open problem in that case.…”

“…In addition, several analytical approaches have been used to obtain time-periodic solutions formally for different types of piecewiselinear dynamical systems with rigid impacts. One can mention Fourier and Green function methods [4,5,8,17,18,19,23,24,25,31,37], modal decomposition [27,38] and sawtooth time transformations [32]. Most of the results obtained for discrete systems concern impacts localized on a single particle, and different types of waves have been constructed.…”

“…In this work, we study the existence and stability of time-periodic oscillations in an infinite chain of linearly coupled impact oscillators reminiscent of a model analyzed in [19,31], for rigid impacts without energy dissipation. We show the existence of exact solutions (i.e.…”