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 functional-differential 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, quasi-periodic 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 $u^3$ $(|u|^pu, p\geq2)$ 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.