Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem

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.Mathematics Subject Classification 2010: 37C25, 39A23 (Primary); 13P15, 34D23, 70F15, 70K05 (Secondary).satisfies the Markus-Yamabe condition (2) and has a 3-periodic orbit.Prior to prove this proposition, we recall the following auxiliary lemma, that is a simplified version of a result given in [12]. Lemma 4. Let G(x; b) = g n (b)x n + g n−1 (b)x n−1 + · · · + g 1 (b)x + g 0 (b) be a family of real polynomials that depend continuously on one real parameter b ∈ B = [b 1 , b 2 ] ⊂ R. Fix J = [x, x] ⊂ R and assume that: (i) There exists b 0 ∈ B such that G(x; b 0 ) has no real roots in J.

Many periodic solutions for a second order cubic periodic differential equation

“…Moreover, this system has integer coefficients. For solving such a system we used an algorithm based on resultants, see for instance [18,11]. We will prove that, beside the 5 solutions that we initially imposed and the (0, 0) solution, the system has other three real solutions.…”

“…For higher degree polynomials, the use of Poincaré-Miranda Theorem helps to decide which of the couples, candidate to be a solution of the system, is an actual solution for it, see [10,11].…”

“…Assume that τ (h * ) = 0 and that there exists θ * ∈ [0, T /j) such that M (θ * ) = 0 and M (θ * ) = 0, where M is given in (12). Then there exists ε > 0 such that for each ε ∈ (−ε, ε), equation (11) has a T -periodic solution u ε with lim ε→0 u ε (0) = ξ(θ * ) and with the property that it is isolated in the set of T -periodic solutions of (11).…”

“…In the sequel we provide an example of perturbation g in system (11) such that the Malkin bifurcation function associated satisfies the hypothesis in (ii) of Theorem 17. In addition, we provide an example of such system whose Malkin bifurcation function does not have any zero, thus the system does not have Tperiodic solutions like in (i) of Theorem 17.…”

Limit Cycles for Piecewise Linear Differential Systems via Poincaré–Miranda Theorem

“…On the other two sides of B 1 or on the boundaries of the other 2 boxes we can use the same approach. The only changes are that n and k vary from one to another, the corresponding upper bound M must be computed and sometimes instead of g j is is convenient to consider e v/5 g j , see [2] for more details.…”