Abstract
This paper gives localization and nonexistence conditions of periodic orbits in some subsets of the state space. Mainly, our approach is based on high‐order extremum conditions, on high‐order tangency conditions of a nonsingular solution of a polynomial system with an algebraic surface, and on some ideas related to algebraically‐dependent polynomials. Examples of the localization analysis of periodic orbits are presented including the Blasius equations, the generalized mass action (GMA) system, and the mathematical model of the chemical reaction with autocatalytic step.
Highlights
The study of periodic orbits has a significant impact on understanding the dynamics of multidimensional systems
We present one sufficient nonexistence condition of periodic orbits stated in terms of sets S+2N, S−2N
We described sets in the state space of a nonlinear system, which do not contain periodic orbits, contain all periodic orbits, or have common points with any periodic orbit
Summary
The study of periodic orbits has a significant impact on understanding the dynamics of multidimensional systems. The main objective of this paper is to find some sets K in Rn in terms of conditions imposed on f and h such that (1) any periodic orbit of (1.1) is contained in K, with K a domain, or (2) any periodic orbit has common points with K; here K is a surface. By using these sets, we propose nonexistence conditions of periodic orbits in some domains of the state space.
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