Periodic orbits of discrete and continuous dynamical systems via Poincar\'{e}-Miranda theorem

TL;DR

This paper introduces a systematic analytical method using the Poincaré-Miranda theorem to identify isolated periodic points in discrete and continuous algebraic dynamical systems, demonstrated through diverse complex examples.

Contribution

The paper develops a novel, systematic approach leveraging the Poincaré-Miranda theorem for locating periodic points in algebraic dynamical systems, applicable to various complex scenarios.

Findings

Identified counterexamples to the Markus-Yamabe conjecture.

Analyzed low periods of a Lotka-Volterra map.

Proved existence of three limit cycles in a piecewise linear vector field.

Abstract

We present a systematic methodology to determine and locate analytically isolated periodic points of discrete and continuous dynamical systems with algebraic nature. We apply this method to a wide range of examples, including a one-parameter family of counterexamples to the discrete Markus-Yamabe conjecture (La Salle conjecture); the study of the low periods of a Lotka-Volterra-type map; the existence of three limit cycles for a piece-wise linear planar vector field; a new counterexample of Kouchnirenko's conjecture; and an alternative proof of the existence of a class of symmetric central configuration of the $(1+4)$-body problem.