Joan Torregrosa

CENTER-FOCUS PROBLEM FOR DISCONTINUOUS PLANAR DIFFERENTIAL EQUATIONS

We study the center-focus problem as well as the number of limit cycles which bifurcate from a weak focus for several families of planar discontinuous ordinary differential equations. Our computations of the return map near the critical point are performed with a new method based on a suitable decomposition of certain one-forms associated with the expression of the system in polar coordinates. This decomposition simplifies all the expressions involved in the procedure. Finally, we apply our results to study a mathematical model of a mechanical problem, the movement of a ball between two elastic walls.

Limit cycles in the Holling-Tanner model

This paper deals with the following question: does the asymptotic stability of the positive equilibrium of the Holling-Tanner model imply it is also globally stable? We will show that the answer to this question is negative. The main tool we use is the computation of Poincare-Lyapunov constants in case a weak focus occurs. In this way we are able to construct an example with two limit cycles.

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.

Bifurcation of limit cycles from a centre in R-4 in resonance 1:N

For every positive integer N ≥ 2 we consider the linear differential centre in ℝ4 with eigenvalues ±i and ±Ni. We perturb this linear centre inside the class of all polynomial differential systems of the form linear plus a homogeneous nonlinearity of degree N, i.e. where every component of F(x) is a linear polynomial plus a homogeneous polynomial of degree N. Then if the displacement function of order ϵ of the perturbed system is not identically zero, we study the maximal number of limit cycles that can bifurcate from the periodic orbits of the linear differential centre.

Exact number of limit cycles for a family of rigid systems

For a given family of planar differential equations it is a very difficult problem to determine an upper bound for the number of its limit cycles. Even when this upper bound is one it is not always an easy problem to distinguish between the case of zero and one limit cycle. This note mainly deals with this second problem for a family of systems with a homogeneous nonlinear part. While the condition that allows us to separate the existence and the nonexistence of limit cycles can be described, it is very intricate.

Some results on homoclinic and heteroclinic connections in planar systems

Consider a family of planar systems depending on two parameters (n, b) and having at most one limit cycle. Assume that the limit cycle disappears at some homoclinic (or heteroclinic) connection when Φ(n, b) = 0. We present a method that allows us to obtain a sequence of explicit algebraic lower and upper bounds for the bifurcation set Φ(n, b) = 0. The method is applied to two quadratic families, one of them is the well-known Bogdanov–Takens system. One of the results that we obtain for this system is the bifurcation curve for small values of n, given by . We obtain the new three terms from purely algebraic calculations, without evaluating Melnikov functions.

Global Classification of a class of Cubic Vector Fields whose canonical regions are period annuli

We study cubic vector fields with inverse radial symmetry, i.e. of the form ẋ = δx - y + ax2 + bxy + cy2 + σ(dx - y)(x2 + y2), ẏ = x + δy + ex2 + fxy + gy2 + σ(x + dy) (x2 + y2), having a center at the origin and at infinity; we shortly call them cubic irs-systems. These systems are known to be Hamiltonian or reversible. Here we provide an improvement of the algorithm that characterizes these systems and we give a new normal form. Our main result is the systematic classification of the global phase portraits of the cubic Hamiltonian irs-systems respecting time (i.e. σ = 1) up to topological and diffeomorphic equivalence. In particular, there are 22 (resp. 14) topologically different global phase portraits for the Hamiltonian (resp. reversible Hamiltonian) irs-systems on the Poincare disc. Finally we illustrate how to generalize our results to polynomial irs-systems of arbitrary degree. In particular, we study the bifurcation diagram of a 1-parameter subfamily of quintic Hamiltonian irs-systems. Moreover, we indicate how to construct a concrete reversible irs-system with a given configuration of singularities respecting their topological type and separatrix connections.

Limit cycles coming from some uniform isochronous centers

Abstract This article is about the weak 16th Hilbert problem, i.e. we analyze how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed inside a class of polynomial differential systems. More precisely, we consider the uniform isochronous centers x ˙ = - y + x 2 ⁢ y ⁢ ( x 2 + y 2 ) n , y ˙ = x + x ⁢ y 2 ⁢ ( x 2 + y 2 ) n ,

GLOBAL PHASE PORTRAITS OF SOME REVERSIBLE CUBIC CENTERS WITH COLLINEAR OR INFINITELY MANY SINGULARITIES

\dot{x}=-y+x^{2}y(x^{2}+y^{2})^{n},\quad\dot{y}=x+xy^{2}(x^{2}+y^{2})^{n},

Limit cycles bifurcating from a 2-dimensional isochronous torus in R 3

of degree 2 ⁢ n + 3