Birth of limit cycles for a class of continuous and discontinuous differential systems in (d+ 2)–dimension

Abstract

Abstract. The orbits of the reversible differential systemẋ = −y,ẏ = x,ż = 0, with x, y ∈ R and z ∈ R d , are periodic with the exception of the equilibrium points (0, 0, z). We compute the maximum number of limit cycles which bifurcate from the periodic orbits of the systemẋ = −y,ẏ = x,ż = 0, using the averaging theory of first order, when this system is perturbed, first inside the class of all polynomial differential systems of degree n, and second inside the class of all discontinuous piecewise polynomial differential systems of degree n with two pieces, one in y > 0 and the other in y < 0. In the first case this maximum number is n d (n − 1)/2, and in the second is n d+1 .

Limit cycles of piecewise polynomial perturbations of higher dimensional linear differential systems

“…Theorem 3 generalizes the particular case m = d of [26]. Comparing itens (a) and (b) of Theorem 3, we can easily check that N 2 (m, n, φ) > N 1 (m, n, φ) for every 0 ≤ m ≤ d, n ∈ N, and φ ∈ (0, 2π) \ {π}.…”

“…Then, in [20,24], the averaging theory was extended up to order 2 for detecting periodic orbits of discontinuous piecewise smooth differential systems. Some applications of these results can be found in [26,29]. Finally, in [16,22], the averaging theory was developed at any order for a class of discontinuous piecewise smooth systems.…”

“…The first studies on this subject considered smooth differential systems and, since then, many contributions have been made in this direction (see [15] and the references therein). The study of limit cycles has also been considered for continuous (see, for instance, [4,23,25]) and discontinuous piecewise smooth differential systems (see, for instance, [11,14,19,20,26]). Most of them are concentrated on planar piecewise differential systems.…”

Periodic solutions of discontinuous second order differential systems

“…where F 1 , F 2 , R 1 , R 2 and h are continuous functions, locally Lipschitz in the variable x, T -periodic in the variable t, and h is a C 1 function having 0 as a regular value. The results stated in [19] have been extensively used, see for instance the works [16,17,26,21,22]. In this paper we focus on the development and improvement of the averaging theory for studying periodic solutions of a much bigger class of discontinuous differential systems than in (1).…”

“…where F 1 , F 2 , R 1 , R 2 and h are continuous functions, locally Lipschitz in the variable x, T -periodic in the variable t, and h is a C 1 function having 0 as a regular value. The results stated in [19] have been extensively used, see for instance the works [16,17,26,21,22].…”