Abstract
The paper studies the existence problem of periodic solutions of the nonlinear dynamical systems in the singular case. We prove a certain generalization of the Andronov-Hopf theorem. This generalization is based on an application of the theorem on a modified p-factor operator. It also uses some other results and constructions of the p-regularity theory. Moreover, we prove theorems on the solution’s uniqueness. We illustrate our results by the example of a nonlinear dynamical system of ordinary differential equations. Our purpose is to find periodic solutions of such system with fixed period 2π. This is a new research in relation to previous work, where the authors were looking for periodic solutions with period near 2π.
Highlights
1 Introduction In this paper we study the structure of periodic solutions of dynamical systems and bifurcation problems associated with such systems, i.e., we consider a nonlinear system of differential equations of the form u = f (μ, u), u( ) = u(τ ), ( )
Our article presents some generalization of Andronov-Hopf theorem to solve a similar problem but in a different way. It is a continuation of work by Medak and Tret’yakov [ ], where the authors presented a different modification of the theorem, which gives an effective method to analyze the existence of periodic solutions of nonlinear dynamical systems
10 Conclusion The paper is devoted to the problem of the existence of periodic solutions of a dynamical system which can be investigated by means of p-regularity theory
Summary
It turns out that the apparatus of p-regularity theory gives us the ability to construct a wide class of p-factor operators, by means of which one can describe the tangent cone to the sets of solutions and get the solutions (see [ , ]) We will call such operators modified or generalized. In this paper we prove a new theorem on the modified p-factor operator which is some generalization of the Andronov-Hopf theorem. The new theorems on the solution’s uniqueness will be proved too These results we can consider as a contribution to and a novelty in nonlinear differential equations theory that we represent in our paper. The notion of regularity is generalized to the notion of the so-called p-regularity
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