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Interactive optimization as a tool for finding the complex periodic solutions in nonlinear dynamics

I n t r o d u c t i o n. We consider essentially nonlinear dynamical systems with the ability to implement a chaotic behavior and deterministic solutions of various kinds. Among the deterministic solutions, we will highlight a variety of periodic solutions of different periods. Problems of control of dynamic regimes in such systems discussed in [1]. This work is devoted to numerical algorithms for constructing and analyzing the stability of periodic solutions of strongly nonlinear dynamical systems. P r o b l e m S t a t e m e n t. We will consider the strongly nonlinear system, which places no restrictions on the value of the individual components. In the framework of this approach we can analyze the linear and quasi-linear system, but the focus will be given to an essentially nonlinear systems of general form. The only requirement that we make to a dynamic system is the ability to construct numerical solutions of the Cauchy problem with the required precision. We will use a general approach to the problem of constructing periodic solutions of nonlinear systems of ordinary differential equations. This approach H. Poincare [2] formulated as follows: Suppose that