Localization in finite asymmetric vibro-impact chains

TL;DR

This paper derives exact analytic solutions for localized discrete breathers in a finite asymmetric vibro-impact chain, analyzing their stability and bifurcations in both conservative and forced-damped regimes.

Contribution

It introduces a novel vibro-impact chain model with asymmetry allowing explicit breather solutions and stability analysis using linear algebra methods.

Findings

Exact analytic solutions for asymmetric discrete breathers.

Identification of bifurcations leading to loss of stability.

Explicit stability analysis via monodromy matrix.

Abstract

We explore the dynamics of strongly localized periodic solutions (discrete solitons, or discrete breathers) in a finite one-dimensional chain of asymmetric vibro-impact oscillators. The model involves a parabolic on-site potential with asymmetric rigid constraints (the displacement domain of each particle is finite), and a linear nearest-neighbor coupling. When the particle approaches the constraint, it undergoes an impact (not necessarily elastic), that satisfies Newton impact law. Nonlinearity of the system stems from the impacts; their possible non-elasticity is the sole source of damping in the system. We demonstrate that this vibro-impact model allows derivation of exact analytic solutions for the asymmetric discrete breathers, both in conservative and forced-damped settings. The asymmetry makes two types of breathers possible: breathers that impact both or only one constraint. Transition between these two types of the breathers corresponds to a grazing bifurcation. Special character of the nonlinearity permits explicit derivation of a monodromy matrix. Therefore, the stability of the obtained breather solutions can be exactly studied in the framework of simple methods of linear algebra, and with rather moderate computational efforts. All three generic scenarios of the loss of stability (pitchfork, Neimark-Sacker and period doubling bifurcations) are observed.