Stability analysis of dynamical regimes in nonlinear systems with discrete symmetries

Abstract

We present a theorem that allows to simplify linear stability analysis of periodic and quasiperiodic nonlinear regimes in N -particle mechanical systems (both conservative and dissipative) with different kinds of discrete symmetry. This theorem suggests a decomposition of the linearized system arising in the standard stability analysis into a number of subsystems whose dimensions can be considerably less than that of the full system. As an example of such simplification, we discuss the stability of bushes of modes (invariant manifolds) for the Fermi-Pasta-Ulam chains and prove another theorem about the maximal dimension of the above mentioned subsystems.

Nonlinear vibrational modes in graphene: group-theoretical results

“…Each Rosenberg mode describes a periodic dynamical regime. In [9][10][11][12][13][14], the bushes of vibrational modes were studied in various nonlinear systems of different physical nature with some point and space symmetry groups.…”

Delocalized periodic vibrations in nonlinear LC and LCR electrical chains

“…As was shown in [18], this structure of the matrix J = J(t), obtained after basis transformation producing the form (21) of the representation Γ, leads to splitting (decomposition) of the original variational systemδ = J(t)δ into a number of independent subsystems whose dimensions can be considerably smaller than that of the original variational system. Namely, n j identical subsystems of dimension m j correspond to every block D j and, therefore, to the irrep Γ j .…”

“…(32) represents a system of linear algebraic equations for unknown vectors φ i , i = 1..n. Traditionally, basis vectors of an irreducible representation are found by means of projection operators (see, for example [31]). However, it is more convenient for our purpose to obtain these vectors using the straightforward method [32] based on the definition (32) of matrix representation (some details of this method can be found in [18]). Obviously, it is sufficient to use equations (32) only for the generators of the group G.…”

Dissipative Discrete System with Nearest-Neighbor Interaction for the Nonlinear Electrical Lattice

“…in modern nonlinear science the concept of bushes of normal modes could be applied. The phenomenon observe in Figure 7 can be also explained by the theory of "bushes" of nonlinear normal modes [35][36][37]. Since the symmetry-determined bushes are valid for any of monatomic chains and, in some sense, they can be applied to multiatomic chains as well, one can describe these phenomena as bushes.…”

“…As an indivisible nonlinear object, the bush exists because of force interactions between the modes contained in it. Apparently, bushes of modes play an important role in many physical phenomena of current interest [35][36][37].…”