The main purpose of this paper is to present a systematic methodology to determine and locate analytically isolated periodic points of discrete and continuous dynamical systems with algebraic nature. This method is applied to a wide range of examples, including a one-parameter family of counterexamples to the discrete Markus-Yamabe conjecture (La Salle conjecture). The authors consider: the study of the low periods of a Lotka-Volterra-type map; the existence of three limit cycles for a piecewise linear planar vector field; a new counterexample to Kouchnirenko’s conjecture; an alternative proof of the existence of a class of symmetric central configuration of the \((1 + 4)\)-body problem.
More precisely, they start giving a new degree-6 counterexample to Kouchnirenko’s conjecture to illustrate the validity and performance of their approach. In Section 3, they prove the existence of a 1-parameter family of rational counterexamples to a conjecture of La Salle (also known as discrete Markus-Yamabe conjecture) that extends the results of A. Cima et al. [Publ. Mat., Barc. 2014, 167–178 (2014; Zbl 1308.39015)], providing also an alternative proof.
In Section 4 they show the existence of exactly two 5-periodic orbits and three 6-periodic orbits in a certain region for a Lotka-Volterra-type map, correcting and complementing some results that appear in the literature.
In Section 5 another example of planar piecewise linear differential system with two zones having 3-limit cycles is provided.
Finally, in Section 6 the Poincaré-Miranda theorem is used to give an alternative proof of the existence of a type of symmetric central configuration of the \((1 + 4)\)-body problem.