We investigate the existence and stability of discrete breathers in a chain of masses connected by linear springs and subjected to vibro-impact on-site potentials. The latter are comprised of harmonic springs and rigid constraints limiting the possible motion of the masses. Local dissipation is introduced through a non-unit restitution coefficient characterizing the impacts. The system is excited by uniform time-periodic forcing. The present work is aimed to study the existence and stability of similar breathers in the space of parameters, if additional harmonic potentials are introduced. Existence-stability patterns of the breathers in the parameter space and possible bifurcation scenarios are investigated analytically and numerically. In particular, it is shown that the addition of harmonic on-site potential can substantially extend the stability domain, at least close to the anti-continuum limit. This result can be treated as an increase in the robustness of the breather from the perspective of possible practical applications.

Variable-amplitude oscillatory shear tests are emerging as powerful tools to investigate and quantify the nonlinear rheology of amorphous solids, complex fluids and biological materials. Quite a few recent experimental and atomistic simulation studies demonstrated that at low shear amplitudes, an amorphous solid settles into an amplitude-and initial conditions-dependent dissipative limit cycle, in which back-and-forth localized particle rearrangements periodically bring the system to the same state. At sufficiently large shear amplitudes, the amorphous system loses memory of the initial conditions, exhibits chaotic particle motions accompanied by diffusive behavior and settles into a stochastic steady-state. The two regimes are separated by a transition amplitude, possibly characterized by some critical-like features. Here we argue that these observations support some of the physical assumptions embodied in the nonequilibrium thermodynamic, internal-variables based, Shear-Transformation-Zone model of amorphous visco-plasticity; most notably that "flow defects" in amorphous solids are characterized by internal states between which they can make transitions, and that structural evolution is driven by dissipation associated with plastic deformation. We present a rather extensive theoretical analysis of the thermodynamic Shear-Transformation-Zone model for a variable-amplitude oscillatory shear protocol, highlighting its success in accounting for various experimental and simulational observations, as well as its limitations. Our results offer a continuum-level theoretical framework for interpreting the variable-amplitude oscillatory shear response of amorphous solids and may promote additional developments. I. BACKGROUND AND MOTIVATIONThe application of oscillatory shear deformation of the form γ(t) = γ 0 sin(ω t) -where γ(t) is the time-dependent shear strain, t is time, γ 0 is the shear amplitude and ω is the oscillations frequency -offers an important protocol to probe the rheological properties of a broad range of physical systems, including amorphous solids, complex fluids and biological materials. The most well-developed and well-documented rheological test in this context focusses on the linear viscoelastic response, where the amplitude γ 0 is very small and the frequency ω is systematically varied. In this case, the steady-state linear stress response is fully characterized by a single frequencydependent complex function G * (ω) = G ′ (ω) + iG ′′ (ω), where the storage (shear) modulus G ′ (ω) quantifies the linear elastic response and the loss modulus G ′′ (ω) quantifies the linear viscous response.A complementary protocol which sheds light on nonlinear material rheology, and the transition between linear and nonlinear rheologies, is obtained by fixing the frequency ω and systematically varying the amplitude γ 0 , within and well beyond the linear response regime. Such variable-amplitude oscillatory shear tests applied to amorphous solids have been the focus of quite a few recent simulational and experimenta...

We consider a discrete dynamical system with internal degrees of freedom (DOF). Due to the symmetry between the internal DOFs, certain internal modes cannot be excited by external forcing (in a case of linear interactions) and thus are considered "hidden". If such a system is weakly asymmetric, the internal modes remain approximately "hidden" from the external excitation, given that small damping is taken into account. However, already in the case of weak cubic nonlinearity, these hidden modes can be excited, even as the exact symmetry is preserved. This excitation occurs through parametric resonance. Floquet analysis reveals instability patterns for the explored modes. To perform this analysis with the required accuracy, we suggest a special method for obtaining the Fourier series of the unperturbed solution for the nonlinear normal mode. This method does not require explicit integration of the arising quadratures. Instead, it employs expansion of the solution at the stage of the implicit quadrature in terms of Chebyshev polynomials. The emerging implicit equations are solved by using a fixed-point iteration scheme. Poincar\'{e} sections help to clarify the correspondence between the loss of stability of the modes and the global structure of the dynamical flow. In particular, the conditions for intensive energy exchange in the system are characterized

Local configurational symmetry in lattice structures may give rise to stationary, compact solutions, even in the absence of disorder and nonlinearity. These compact solutions are related to the existence of flat dispersion curves (bands). Nonlinearity can destabilize such compactons. One common flat-band-generating system is the one-dimensional cross-stitch model, in which compactons were shown to exist for the photonic lattice with Kerr nonlinearity. The compactons exist there already in the linear regime and are not generally destructed by that nonlinearity. Smooth nonlinearity of this kind does not permit performing complete stability analysis for this chain. We consider a discrete mechanical system with flat dispersion bands, in which the nonlinearity exists due to impact constraints. In this case, one can use the concept of the saltation matrix for the analytic construction of the monodromy matrix. Besides, we consider a smooth nonlinear lattice with linearly connected massless boxes, each containing two symmetric anharmonic oscillators. In this model, the flat bands and discrete compactons also readily emerge. This system also permits performing comprehensive stability analysis, at least in the anticontinuum limit, due to the reduced number of degrees of freedom. In both systems, there exist two types of localization. The first one is the complete localization, and the second one is the more common exponential localization. The latter type is associated with discrete breathers (DBs). Two principal mechanisms for the loss of stability are revealed. The first one is the possible internal instability of the symmetric and/or antisymmetric solution in the individual unit cell of the chain. One can interpret this instability pattern as internal resonance between the compacton and the DB. The other mechanism is global instability related to resonance of the stationary solution with the propagation frequencies. Different instability mechanisms lead to different bifurcations at the stability threshold.

We consider a system of two linear and linearly coupled oscillators with ideal impact constraints. Primary resonant energy exchange is investigated by analysis of the slow flow using the action-angle (AA) formalism. Exact inversion of the action-energy dependence for the linear oscillator with impact constraints is not possible. This difficulty, typical for many models of nonlinear oscillators, is circumvented by matching the asymptotic expansions for the linear and impact limits. The obtained energy-action relation enables the complete analysis of the slow flow and the accurate description of the critical delocalization transition. The transition from the localization regime to the energy-exchange regime is captured by prediction of the critical coupling value. Accurate prediction of the delocalization transition requires a detailed account of the coupling energy with appropriate redefinition and optimization of the limiting phase trajectory on the resonant manifold.This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.