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In particular, one-dimensional chains of linearly coupled nonlinear oscillators are addressed by means of nonlinear maps. Chains of oscillators with cubic nonlinearities have been studied in, through numerical and asymptotic approaches, respectively.īeing interested in dynamical phenomena with the same length scale of the interoscillator distance, discrete nonlinear systems are considered in this work, whose main goal is to analytically investigate the modification of the boundary of the linear propagation/attenuation zones due to the nonlinearities. The existence of transitions from soliton-like motions to spatially and temporally disordered motions due to a sudden excitation has been shown relying on a modified Toda lattice model. In an array of elastic oscillators coupled through buckling sensitive elastica has been addressed both numerically and experimentally. Two different asymptotic approaches have been devised for studying standing (stop-band) and traveling (pass-bands) waves amplitude dependent frequencies bounding nonlinear propagation and attenuation zones have been found. Monocoupled periodic systems of infinite extent with material nonlinearities have been addressed in. In contrast to the vast literature concerning linear periodic structures, few works have so far appeared in the mechanical context on the dynamics of nonlinear periodic systems. Moreover, the natural frequencies fall within pass-bands. It is well known that the dynamics of linear multi-coupled periodic systems is governed by frequency intervals or bands where disturbances propagate harmonically (pass-bands), decay (stop-bands) or propagate harmonically with attenuation (complex-bands). In the latter, periodically reinforced structures as well as trusses, are frequently used in aerospace, naval, and civil engineering and can be conveniently modeled as periodic systems. One-dimensional chains of coupled oscillators are suitable for modeling a number of physical systems arising in different scientific contexts such as condensed-matter physics, optics, chemistry and mechanics. The analytical findings concerning the propagation properties are then compared with numerical results obtained through nonlinear map iteration. Also in this case, where a 4D real map governs the wave transmission, the nonlinear pass- and stop-bands for periodic orbits are analytically determined by extending the 2D map analysis. Then, equivalent chains of nonlinear oscillators in complex domain are tackled. Pass- and stop-band regions of the mono-coupled periodic system are analytically determined for period- q orbits as they are governed by the eigenvalues of the linearized 2D map arising from linear stability analysis of periodic orbits. Mechanical models of chains of linearly coupled nonlinear oscillators are investigated. Thus, wave propagation becomes synonymous of stability: finding regions of propagating wave solutions is equivalent to finding regions of linearly stable map solutions. In this realm, the governing difference equations are regarded as symplectic nonlinear transformations relating the amplitudes in adjacent chain sites ( n, n + 1) thereby considering a dynamical system where the location index n plays the role of the discrete time.
Trackingtime waveapp free#
Free wave propagation properties in one-dimensional chains of nonlinear oscillators are investigated by means of nonlinear maps.