Nonlinear dynamics of hidden modes in a system with internal symmetry

Abstract

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

Flat bands and compactons in mechanical lattices

“…18 (bottom) shows the long-time history confirming the persistence of the energyexchange phenomenon after all transients are gone. This energy exchange between modes is similar to what is observed (for other parameter values) on a y-y Poincaré section for a single element [41]. Figure 19 shows the flow close to a global bifurcation where KAM (Kolmogorov-Arnold-Moser) islands of the symmetric and antisymmetric modes "collide", corresponding to the aforementioned significant (1:2 resonance for both modes) energy exchange (see [41]).…”

“…Asymptotic analysis of Eq. (66), resulting in the identification of the number of emerging instability tongues, derivation of starting points at the zero-amplitude limit for numerical instability-tongue boundaries calculation, as well as the derivation of leading-order asymptotic expansions for the boundaries of the first two instability tongues, using both Mathieu and higher-order approximations, can be found in [41].…”

“…(52), one obtains the (algebraic) equation of motion for the displacement of the box, as follows: Additional mathematical manipulations, described in [41], produce the explicit Hill form of the linear stability equation for the purely antisymmetric mode:…”

“…17. The single-element analysis presented here is reproduced in part from [41], for better understanding of the dynamics in the context of a chain. This owes to the fact that for a chain, rigorous stability analysis is not within reach, other than when understanding the single element case as representative of the anti-continuum limit for a chain, taking k/k 0 1.…”

“…3-4 present the complete stability analysis and numerical verification of the results for the non-smooth model; Sec. 5 addresses the chain of massless boxes with internal nonlinear oscillators, related to the system analyzed in [41]. Section 6 establishes the existence of stable compacton solutions for the chain by direct numerical integration, and is followed by Sec.…”