Multifunctional variational method for description of evolution and dynamics of dissipative structures in nonequilibrium systems

A variational principle based on the introduction of a vector functional, each component whereof has its extremum with respect to variation of only one (or a group of) macroscopic parameter(s) of a system, is presented. The suggested method greatly facilitates the description of the characteristic parameters of stationary and time-dependent complex inhomogeneous macroscopic states occurring in nonequilibrium systems. The fruitfulness and simplicity of the presented method are illustrated by analysis of bifurcation types and by studying the shape, stability, and evolution of spike strata and autosolitions. A nonlinear theory of pulsating spike strata and autosolitons of large amplitude is developed. It is shown that the variations of the characteristic parameters of pulsating strata and autosolitons are relaxational spike auto-oscillations which may be of periodical as well as of apparently chaotic character. A simple method of analysis of quasiharmonic states is developed and, for a concrete model, it is shown that supercritical-solution bifurcations take place only if parameters of the system meet very rigorous requirements, i.e., it is confirmed that instability of the system’s homogeneous states leads, as a rule, to abrupt formation of large-amplitude structures. The proposed variational principle is shown to be useful for deriving a set of ordinary differential equations, which describe in a simple way the interaction between autosolitons and turbulence in active distributed media.