Stability of nonlinear normal modes in the Fermi-Pasta-Ulam ß chain in the thermodynamic limit.

All possible symmetry-determined nonlinear normal modes (also called simple periodic orbits, one-mode solutions, etc.) in both hard and soft Fermi-Pasta-Ulam ß chains are discussed. A general method for studying their stability in the thermodynamic limit as well as its application for each of the above nonlinear normal modes are presented.

Discrete breathers on symmetry-determined invariant manifolds for scalar models on the plane square lattice.

A group-theoretical approach for studying localized periodic and quasiperiodic vibrations in two- and three-dimensional lattice dynamical models is developed. This approach is demonstrated for the scalar models on the plane square lattice. The symmetry-determined invariant manifolds admitting existence of localized vibrations are found, and some types of discrete breathers are constructed on these manifolds. A general method using the apparatus of matrix representations of symmetry groups to simplify the standard linear stability analysis is discussed. This method allows one to decompose the corresponding system of linear differential equations with time-dependent coefficients into a number of independent subsystems whose dimensions are less than the full dimension of the considered system.

Quasibreathers as a generalization of the concept of discrete breathers.

The majority of dynamical objects, which demonstrate energy localization in nonlinear lattices, represent quasibreathers rather than strictly time-periodic discrete breathers since, as a rule, there exist certain deviations in vibrational frequencies of the individual particles exceeding the possible numerical errors. We illustrate this idea with the James breathers in the K2-K3-K4 chain and with quasibreathers in the K4 chain. For the latter case, a rigorous investigation of existence and stability of the breathers and quasibreathers is presented. In particular, it is proved that they are stable up to a certain strength of the intersite part of the potential with respect to its on-site part. We conjecture that quasibreathers play a fundamental role in the problem of energy localization in more realistic nonlinear lattices, as well. The difference between breathers and quasibreathers can be characterized by the mean square deviation of the frequencies of individual particles.

Stability analysis of dynamical regimes in nonlinear systems with discrete symmetries.

We present a theorem that allows one to simplify the linear stability analysis of periodic and quasiperiodic nonlinear regimes in N-particle mechanical systems 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 the dimension of the full system. As an example of such a 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.

Three-dimensional chaotic flows with discrete symmetries.

There are a number of well-known three-dimensional flows with quadratic nonlinearities, which demonstrate a chaotic behavior. The most popular among them are Lorenz and Rössler systems. Using an exhaustive computer search, J. Sprott found 19 examples of chaotic flows with either five terms and two quadratic nonlinearities or six terms and one nonlinearity [Phys. Rev. E 50, R647 (1994)]. In contrast to this approach, we use symmetry-related considerations to construct types of chaotic flows with an arbitrary dimension. The discussion is based on our previous work devoted to nonlinear dynamics of the physical systems with discrete symmetries [see Physica D 117, 43 (1998), etc.]. Here, we present all possible chaotic flows with quadratic nonlinearities which are invariant under the action of 32 point groups of crystallographic symmetry. These systems demonstrate a typical chaotic behavior as well as general dynamical properties of nonlinear systems with discrete symmetries. In particular, we found a dynamical system with the point symmetry group D2 which seems to be more simple and more elegant than those by Lorenz and Rössler.