Application of a computer mathematics system in problems of applied nonlinear dynamics

Mathematical formalization of various natural processes leads to models that
are described by nonlinear differential equations (ordinary, in partial derivatives and functionaldifferential equations) or nonlinear integral equations.
Their research takes place within the framework of applied non-linear dynamics. The issues
of stability, bifurcation of solutions, the emergence of spatially inhomogeneous structures, quasiperiodic solutions, etc. are considered. Various theories, methods and algorithms are used (for
example, the theory of bifurcation of vector fields, the theory of central manifolds, the theory
of normal forms, etc.). An important and relevant aspect is the use of computer mathematics
systems.

In the article, using the Wolfram Mathematica package, the nonlinear (quasi-linear)
functional-differential equations of the parabolic type with transformation of spatial variables,
which are simulating real physics experiments in nonlinear optical systems with Kerry
nonlinearity, in which the transformation of a field in a two-dimensional feedback loop leads
to the emergence of spatially heterogeneous, rotating and other structures, are investigated
The conditions under which new structures appear depend on several system parameters: the
diffusion coefficient of the medium, the intensity of the signal source, the transformation in the
feedback loop (for example, rotation, compression-stretching). For local analysis of structures and
description of scenarios of their development asymptotic methods of research of local dynamics
of solutions of functional-differential equations with small diffusion parameter, parameter of
intensity and parameters of transformation of coordinates are used. The numerical solution
and visualization of the results for various parameter values are of interest.Note that various
models containing at least cubic nonlinearity with respect to the desired function of the form u³
(|u| ^pu, p ≥ 2) are used to simulate the formation of rotating structures, travelling waves, vortices.

The importance of studying equations with low diffusion and other parameters is due to the
modern problems of searching for innovative methods of storing, transmitting and processing
information, modern issues of nanotechnology.

Keywords: applied nonlinear dynamics, functional differential equations, bifurcation, stationary
solution, Wolfram Mathematica.