We give a method to obtain formal normal forms of reversible equivariant vector fields. The procedure we present is based on the classical method of normal forms combined with tools from invariant theory. Normal forms of two classes of resonant cases are presented, both with linearization having a 2-dimensional nilpotent part and a semisimple part with purely imaginary eigenvalues.

Generic bifurcation theory was classically well developed for smooth differential systems, establishing results for k-parameter families of planar vector fields. In the present study we focus on a qualitative analysis of 2-parameter families, Z α,β , of planar Filippov systems assuming that Z 0,0 presents a codimension-two minimal set. Such object, named elementary simple two-fold cycle, is characterized by a regular trajectory connecting a visible two-fold singularity to itself, for which the second derivative of the first return map is nonvanishing. We analyzed the codimension-two scenario through the exhibition of its bifurcation diagram.2010 Mathematics Subject Classification. 34A36, 34C23, 37G15.

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 .

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The averaging theory has been extensively employed for studying periodic solutions of smooth and nonsmooth differential systems. Here, we extend the averaging theory for studying periodic solutions a class of regularly perturbed non-autonomous n-dimensional discontinuous piecewise smooth differential system. As a fundamental hypothesis, it is assumed that the unperturbed system has a manifold Z ⊂ R n of periodic solutions satisfying dim(Z) < n. Then, we apply this result to study limit cycles bifurcating from periodic solutions of linear differential systems, x = M x, when they are perturbed inside a class of discontinuous piecewise polynomial differential systems with two zones. More precisely, we study the periodic solutions of the following differential system x = M x + εF n 1 (x) + ε 2 F n 2 (x), in R d+2 where ε is a small parameter, M is a (d+2)×(d+2) matrix having one pair of pure imaginary conjugate eigenvalues, m zeros eigenvalues, and d − m non-zero real eigenvalues.

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In this paper we adapt the method of [P. H. Baptistelli, M. Manoel and I. O. Zeli. Normal form theory for reversible equivariant vector fields. Bull. Braz. Math. Soc., New Series 47 (2016), no. 3, 935-954] to obtain normal forms of a class of smooth bireversible vector fields. These are vector fields reversible under the action of two linear involution and whose linearization has a nilpotent part and a semisimple part with purely imaginary eigenvalues. We show that these can be put formally in normal form preserving the reversing symmetries and their linearization. The approach we use is based on an algebraic structure of the set of this type of vector fields. Although this can lead to extensive calculations in some cases, it is in general a simple and algorithmic way to compute the normal forms. We present some examples, which are Hamiltonian systems without resonance for one case and other cases with certain resonances.2010 Mathematics Subject Classification. 7C80, 34C20, 13A50.