New criterions on nonexistence of periodic orbits of planar dynamical systems and their applications

TL;DR

This paper introduces new criteria for determining the nonexistence of periodic orbits in certain planar dynamical systems, demonstrating their applicability through examples and analyzing the local and global phase structures.

Contribution

It presents novel criteria for ruling out periodic orbits in planar systems, extending classical results and applying them to classify phase portraits and local equilibria.

Findings

New criteria effectively determine nonexistence of periodic orbits

Classified global phase portraits of specific planar systems

Revealed incompleteness in classical local topological classification

Abstract

Characterizing existence or not of periodic orbit is a classical problem and it has both theoretical importance and many real applications. Here, several new criterions on nonexistence of periodic orbits of the planar dynamical system $\dot x=y,~\dot y=-g(x)-f(x,y)y$ are obtained in this paper, and by examples showing that these criterions are applicable, but the known ones are invalid to them. Based on these criterions, we further characterize the local topological structures of its equilibrium, which also show that one of the classical results by A.F. Andreev [Amer. Math. Soc. Transl. 8 (1958), 183--207] on local topological classification of the degenerate equilibrium is incomplete. Finally, as another application of these results, we classify the global phase portraits of a planar differential system, which comes from the third question in the list of the 33 questions posed by A. Gasull and also from a mechanical oscillator under suitable restriction to its parameters.