Valeri Smirnov
ul. Kosygina 4, Moscow, 119991 Russia
Semenov Institute of Chemical Physics RAS
Publications:
Smirnov V. V., Manevitch L. I.
Complex Envelope Variable Approximation in Nonlinear Dynamics
2020, Vol. 16, no. 3, pp. 491515
Abstract
We present the complex envelope variable approximation (CEVA) as a useful and compact
method for analysis of essentially nonlinear dynamical systems. The basic idea is that the
introduction of complex variables, which are analogues of the creation and annihilation operators
in quantum mechanics, considerably simplifies the analysis of a number of nonlinear dynamical
systems. The first stage of the procedure, in fact, does not require any additional assumptions,
except for the proposition of the existence of a singlefrequency stationary solution. This allows
us to study both the stationary and nonstationary dynamics even in the cases when there are no
small parameters in the initial problem. In particular, the CEVA method provides an analysis of
nonlinear normal modes and their resonant interactions in discrete systems for a wide range of
oscillation amplitudes. The dispersion relations depending on the oscillation amplitudes can be
obtained in analytical form for both the conservative and the dissipative nonlinear lattices in the
framework of the mainorder approximation. In order to analyze the nonstationary dynamical
processes, we suggest a new notion — the “slow” Hamiltonian, which allows us to generate the
nonstationary equations in the slow time scale. The limiting phase trajectory, the bifurcations of
which determine such processes as the energy localization in the nonlinear chains or the escape
from the potential well under the action of external forces, can be also analyzed in the CEVA.
A number of complex problems were studied earlier in the framework of various modifications
of the method, but the accurate formulation of the CEVA with the stepbystep illustration is
described here for the first time. In this paper we formulate the CEVA’s formalism and give
some nontrivial examples of its application.

Smirnov V. V., Kovaleva M. A., Manevitch L. I.
Nonlinear Dynamics of Torsion Lattices
2018, Vol. 14, no. 2, pp. 179193
Abstract
We present an analysis of torsion oscillations in quasionedimensional lattices with periodic potentials of the nearest neighbor interaction. A onedimensional chain of point dipoles (spins) under an external field and without the latter is the simplest realization of such a system. We obtained dispersion relations for the nonlinear normal modes for a wide range of oscillation amplitudes and wave numbers. The features of the short wavelength part of the spectrum at largeamplitude oscillations are discussed. The problem of localized excitations near the edges of the spectrum is studied by the asymptotic method. We show that the localized oscillations (breathers) appear near the long wavelength edge, while the short wavelength edge of the spectrum contains only dark solitons. The continuum limit of the dynamic equations leads to a generalization of the nonlinear Schrödinger equation and can be considered as a complex representation of the sineGordon equation.

Kovaleva M., Smirnov V. V., Manevitch L. I.
Stationary and nonstationary dynamics of the system of two harmonically coupled pendulums
2017, Vol. 13, No. 1, pp. 105115
Abstract
An analysis is presented of the nonlinear dynamics of harmonically coupled pendulums without restrictions to oscillation amplitudes. This is a basic model in many areas of mechanics and physics (paraffin crystals, DNA molecules etc.). Stationary solutions of equations of motion corresponding to nonlinear normal modes (NNMs) are obtained. The inversion of the NNM frequencies with increasing oscillation amplitude is found. An essentially nonstationary process of the resonant energy exchange is described in terms of limiting phase trajectories (LPTs), for which an effective analytic representation is obtained in slow timescale. Explicit expressions of threshold values of dimensionless parameters are found which correspond to the instability of NNMs and to the transition (in parametric space) from the full energy exchange between the pendulums to the localization of energy. The analytic results obtained are verified by analysis of the Poincar´e sections describing evolution of the initial system.
