Leonid Manevitch
ul. Kosygina 4, Moscow, 119991 Russia
N. N. Semenov Institute of Chemical Physics, Russian Academy of Sciences
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.

Kevorkov S. S., Khamidullin R. K., Koroleva (Kikot) I. P., Smirnov V. V., Gusarova E. B., Manevitch L. I.
Efficiency of a ThreeParticle Energy Sink: Experimental Study and Numerical Simulation
2018, Vol. 14, no. 3, pp. 355366
Abstract
The results of an experimental and numerical investigation of the dynamics of a string
with three uniformly distributed discrete masses are presented. This system can be used as
a resonant energy sink for protecting structural elements from the effects of undesirable dynamic
loads over a wide frequency range. Preliminary studies of the nonlinear dynamics of the system
under consideration showed its high energy capacity. In this paper, we present the results of
an experimental study in which a shaker’s table mounted cantilever beam was being protected.
As a result, the efficiency of the sink was confirmed, and data were also obtained to refine the
mathematical model. It was shown that the experimental data obtained are in good agreement
with the results of computer simulation.

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.

Koroleva (Kikot) I. P., Manevitch L. I.
Oscillatory chain with elastic supports and bending stiffness under conditions close to acoustic vacuum
2016, Vol. 12, No. 3, pp. 311325
Abstract
We present results of analytical and numerical investigation of the nonstationary planar dynamics of a string with uniformly distributed discrete masses without preliminary tension and taking into account the bending stiffness. Each mass is coupled to the ground by lateral springs without tension which have (effectively) a characteristic that is nonlinearizable in the case of planar motion. The most important limiting case corresponding to lowenergy transversal motions is considered taking into account geometrical nonlinearity. Since such excitations are described by approximate equations where cubic elastic forces contribute the most, oscillations take place under conditions close to the acoustic vacuum. We obtain an adequate analytical description of resonant nonstationary processes in the system under consideration, which correspond to an intensive energy exchange between its parts (clusters) in the domain of low frequencies. Conditions of energy localization are given. The analytical results obtained are supported by computer numerical simulations. The system considered may be used as an energy sink of enhanced effectiveness.

Koroleva (Kikot) I. P., Manevitch L. I.
Oscillatory chain with grounding support in conditions of acoustic vacuum
2015, Vol. 11, No. 3, pp. 487502
Abstract
In this work we investigate dynamics of a string with uniformly distributed discrete masses without tension both analytically and numerically. Each mass is also coupled to the ground through lateral spring which provides effect of cubic grounding support. The most important limiting case of lowenergy transversal oscillations is considered accounting for geometric nonlinearity. Since such oscillations are governed by motion equations with purely cubic stiffness nonlinearities, the chain behaves as a nonlinear acoustic vacuum.We obtained adequate analytical description of resonance nonstationary processes in the system which correspond to intensive energy exchange between parts (clusters) of the chain in lowfrequency domain. Conditions of energy localization are given. Obtained analytical results agree well with results of computer simulations. The considered system is shown to be able to be used as very effective energy sink.

Koroleva (Kikot) I. P., Manevitch L. I.
Weakly coupled oscillators in the presence of elactic support in the conditions of acoustic vacuum
2014, Vol. 10, No. 3, pp. 245263
Abstract
A weightless string without preliminary tension with two symmetric discrete masses, which are
influenced by elastic supports with cubic characteristics, is investigated both by numerical and
analytical methods. The most important limit case corresponding to domination of resonance lowenergy
transversal oscillations is considered. Since such oscillations are described by approximate
equations only with cubic terms (without linear ones), the transversal dynamics occurs n the
conditions of acoustic vakuum. If there is no elastic supports nonlinear normal modes of the
system under investigation coincide with (or are close to) those of corresponding linear oscillator
system. However within the presence of elastic supports one of NNM can be unstable, that
causes formation of two another assymmetric modes and a separatrix which divides them. Such dynamical transition which is observed under certain relation between elastic constants of the
string and of the support, relates to stationary resonance dynamics. This transition determines
also a possibility of the second dynamical transition which occurs when the supports contribution
grows. It relates already to nonstationary resonance dynamics when the modal approach turns
out to be inadequate. Effective description of both dynamical transitions can be attained in terms
of weakly interacting oscillators and limiting phase trajectories, corresponding to complete energy
echange between the oscillators.
