M.M. Sadriddinov

Annotation:

The article considers a method for constructing nonlinear projectors for systems of nonlinear differential equations with an analytic right-hand side in a Banach space. The main objective of the study is to develop and provide a theoretical justification for a mathematical framework for constructing nonlinear projectors, to investigate their analytic properties, and to apply the obtained results to an in-depth analysis of the dynamical structure, qualitative characteristics, and local behavior of nonlinear systems of differential equations. The constructed nonlinear projectors make it possible to decompose the phase space into invariant sets corresponding to stable, unstable, and center manifolds. In the proposed method, the spectral decomposition of the linear part of the system is used as the main tool for investigating the dynamical properties of its nonlinear part. The results of the study demonstrate the importance of nonlinear projectors in revealing the geometric structure of the phase space, analyzing stability and instability, and studying the asymptotic behavior of solutions in the neighborhood of an equilibrium point. This method also makes it possible to establish the relationship between the spectral characteristics of the linear part and the dynamical structure of the nonlinear part of the system. In the final section, a specific example is presented, illustrating the algorithm for constructing nonlinear projectors and the practical application of the proposed method.

Keywords: systems of nonlinear differential equations, nonlinear projectors, linear projectors, spectral decomposition, banach space, invariant manifolds, stable, unstable and central manifolds, matrix spectrum, stability theory, analytic function.