Localization in Finite Asymmetric Vibro-Impact Chains

We explore the dynamics of localized periodic solutions (discrete solitons, or discrete breathers) in a finite one-dimensional chain of asymmetric vibro-impact oscillators. The model is intended to simulate dynamical responses of crack arrays, motion of rigid elements between obstacles, as well as the behavior of arrays of microscopic vibro-impact oscillators. The explored chain involves a parabolic on-site potential with asymmetric rigid constraints (the displacement domain of each particle is finite and asymmetric with respect to its equilibrium position) and a linear nearest-neighbor coupling. When the particle approaches the constraint, it undergoes an impact that satisfies the Newton impact law. The restitution coefficient may be less than unity, and it is the only source of damping in the model. Nonlinearity of the system stems from the impact interactions. We demonstrate that this vibro-impact model allows derivation of exact analytical solutions for the asymmetric discrete breathers, in both conservative and forced-damped settings. The asymmetry makes two types of breathers possible: breathers that impact both constraints or only one constraint. Transition between these two types of breathers occurs through a grazing bifurcation. Special character of the nonlinearity permits explicit derivation of the monodromy matrix. Therefore, the stability of the obtained breather solutions can be studied with the desired accuracy in the framework of simple methods of linear algebra, and with rather moderate computational efforts. All three generic scenarios of loss of stability (pitchfork, Neimark-Sacker, and period doubling bifurcations) are observed.