Marco Antonio Teixeira

Limit cycles of the generalized polynomial Liénard differential equations

We apply the averaging theory of first, second and third order to the class of generalized polynomial Lienard differential equations. Our main result shows that for any n , m ≥ 1 there are differential equations of the form ẍ + f ( x ) ẋ + g ( x ) = 0, with f and g polynomials of degree n and m respectively, having at least [( n + m − 1)/2] limit cycles, where [·] denotes the integer part function.

Stability conditions for discontinuous vector fields

Abstract In this work we describe the qualitative behavior of generic discontinuous vector fields around a point p ∈ R 3 .

Higher order averaging theory for finding periodic solutions via Brouwer degree

In this paper we deal with nonlinear differential systems of the form where for i = 0, 1, ..., k, and are continuous functions, and T-periodic in the first variable, D being an open subset of , and e a small parameter. For such differential systems, which do not need to be of class , under convenient assumptions we extend the averaging theory for computing their periodic solutions to k-th order in e. Some applications are also performed.

On the birth of limit cycles for non-smooth dynamical systems

Abstract The main objective of this work is to develop, via Brower degree theory and regularization theory, a variation of the classical averaging method for detecting limit cycles of certain piecewise continuous dynamical systems. In fact, overall results are presented to ensure the existence of limit cycles of such systems. These results may represent new insights in averaging, in particular its relation with non-smooth dynamical systems theory. An application is presented in careful detail.

Original article: Piecewise linear differential systems with two real saddles

In this paper we study piecewise linear differential systems formed by two regions separated by a straight line so that each system has a real saddle point in its region of definition. If both saddles are conveniently situated, they produce a transition flow from a segment of the splitting line to another segment of the same line, and this produces a generalized singular point on the line. This point is a focus or a center and there can be found limit cycles around it. We are going to show that the maximum number of limit cycles that can bifurcate from this focus is two. One of them appears through a Hopf bifurcation and the second when the focus becomes a node by means of the sliding.

Study of Singularities in Nonsmooth Dynamical Systems via Singular Perturbation

In this article we describe some qualitative and geometric aspects of nonsmooth dynamical systems theory around typical singularities. We also establish an interaction between nonsmooth systems and geometric singular perturbation theory. Such systems are represented by discontinuous vector fields on R � , � ≥ 2, where their discontinuity set is a codimension one algebraic variety. By means of a regularization process proceeded by a blow-up technique we are able to bring about some results that bridge the space between discontinuous systems and singularly perturbed smooth systems. We also present an analysis of a subclass of discontinuous vector fields that present transient behavior in the 2-dimensional case, and we dedicate a section to providing sufficient conditions in order for our systems to have local asymptotic stability.

Singularities of reversible vector fields

Abstract For a large class of reversible vector fields on the plane we present all the topological types and their respective normal forms of the symmetric singularities of codimensions 0, 1 and 2.

LOWER BOUNDS FOR THE MAXIMUM NUMBER OF LIMIT CYCLES OF DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS WITH A STRAIGHT LINE OF SEPARATION

In this paper, we provide a lower bound for the maximum number of limit cycles of planar discontinuous piecewise linear differential systems defined in two half-planes separated by a straight line. Here, we only consider nonsliding limit cycles. For those systems, the interior of any limit cycle only contains a unique equilibrium point or a unique sliding segment. Moreover, the linear differential systems that we consider in every half-plane can have either a focus (F), or a node (N), or a saddle (S), these equilibrium points can be real or virtual. Then, we can consider six kinds of planar discontinuous piecewise linear differential systems: FF, FN, FS, NN, NS, SS. We provide for each of these types of discontinuous differential systems examples with two limit cycles.

Limit cycles of some polynomial differential systems in dimension 2, 3 and 4, via averaging theory

In the qualitative study of a differential system it is important to know its limit cycles and their stability. Here through two relevant applications, we show how to study the existence of limit cycles and their stability using the averaging theory. The first application is a 4-dimensional system which is a model arising in synchronization phenomena. Under the natural assumptions of this problem, we can prove the existence of a stable limit cycle. It is known that perturbing the linear center , , up to first order by a family of polynomial differential systems of degree n in , there are perturbed systems with (n − 1) / 2 limit cycles if n is odd, and (n − 2) / 2 limit cycles if n is even. The second application consists in extending this classical result to dimension 3. More precisely, perturbing the system , , , up to first order by a family of polynomial differential systems of degree n in , we can obtain at most n(n − 1) / 2 limit cycles. Moreover, there are such perturbed systems having at least n(n − 1) / 2 limit cycles.

PIECEWISE SMOOTH REVERSIBLE DYNAMICAL SYSTEMS AT A TWO-FOLD SINGULARITY

This paper focuses on the existence of closed orbits around a two-fold singularity of 3D discontinuous systems of the Filippov type in the presence of symmetries.