Complex Envelope Variable Approximation in Nonlinear Dynamics
Received 27 April 2020
2020, Vol. 16, no. 3, pp. 491-515
Author(s): Smirnov V. V., Manevitch L. I.
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 single-frequency 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 main-order 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 step-by-step 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.
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