Chaos and Anderson localization in disordered classical chains: Hertzian versus Fermi-Pasta-Ulam-Tsingou models

Abstract

We numerically investigate the dynamics of an one-dimensional disordered lattice using the Hertzian model, describing a granular chain, and the α + β Fermi-Pasta-Ulam-Tsingou model (FPUT). The most profound difference between the two systems is the discontinuous nonlinearity of the granular chain appearing whenever neighboring particles are detached. We therefore sought to unravel the role of these discontinuities in the destruction of Anderson localization and their influence on the system's chaotic dynamics. Our results show that both models exhibit an energy range where localization coexists with chaos. However, the discontinuous nonlinearity is found to be capable of triggering energy spreading of initially localized modes, at lower energies than the FPUT model. A transition from Anderson localization to chaotic dynamics and energy equipartition is found for the granular chain and is associated with the"propagation" of the discontinuous nonlinearity in the chain. On the contrary, the FPUT chain exhibits an alternate behavior between localized and delocalized chaotic behavior which is strongly dependent on the initial energy of excitation.

Identifying localized and spreading chaos in nonlinear disordered lattices by the Generalized Alignment Index (GALI) method

“…An important outcome of our study was the distinction of the chaotic cases to two different categories of dynamical behaviors, namely cases leading to chaotic localization and cases resulting to chaotic spreading of energy, as was done for example in [60] for a strongly disordered lattice. We showed that, as we move away from the linear system (when energy grows), chaotic dynamics becomes relevant for the DKG model and localized chaos prevails.…”

“…In the presence of nonlinearity the dynamics becomes more complicated as the system's normal modes (NMs) couple and chaos appears. Thus, the interplay of disorder and nonlinearity has attracted extensive attention in theory [12,13,14,15,16,17,18,19,20,21,22,23,24,25], numerical simulations [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,…”

“…Nowadays it is known that wave-packet spreading in nonlinear disordered lattices is a chaotic process resulting to the wave-packet's randomization and chaotization [26,33,15,46,48,53,59,61]. The computation of the maximum Lyapunov Characteristic Exponent (mLCE) [71,72,73] has been used to quantify the degree of chaoticity of disordered lattices [39,46,50,53,58,60], while a detailed analysis of the time evolution of the mLCE's estimator Λ 1 in [48,59,61], showed power law decays Λ 1 ∝ t a Λ , with a Λ ≈ −0.25 (−0.3) and a Λ ≈ −0.37 (−0.46) for the weak (strong) chaos case, respectively for 1D and 2D systems. These behaviors indicate the decline of the systems' chaoticity, which nevertheless dose not shows any signs of a potential crossover to regular dynamics (which is characterized by a Λ ≈ −1), at least up to the computationally achieved (long) integration times.…”

“…We focus our attention on the DKG model as it is computationally easier than the DDNLS system, which typically requires two orders of magnitude more computational time for reaching, with the same accuracy, the same final integration times. We distinguish between localized and extended chaos, something which was not explicitly considered in previous publications, discriminating between energy excitations leading to regular behavior, localized chaos and to delocalized spreading chaotic wave-packets as was done for example in [60].…”

Wave propagation in a strongly disordered 1D phononic lattice supporting rotational waves

“…[11][12][13] where wave-packet spreading was quantified using both analytical and numerical methods. Moreover, many variations of these 1D lattices have been studied extensively in several works including all the regimes from the periodic linear to the disordered nonlinear [10,[18][19][20][21][22][23][24][25].…”