Armengol Gasull

MONODROMY AND STABILITY FOR NILPOTENT CRITICAL POINTS

We give a new and short proof of the characterization of monodromic nilpotent critical points. We also calculate the first generalized Lyapunov constants in order to solve the stability problem. We apply the results to several families of planar systems obtaining necessary and sufficient conditions for having a center. Our method also allows us to generate limit cycles from the origin.

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.

The focus-centre problem for a type of degenerate system

We consider differential systems in the plane defined by the sum of two homogeneous vector fields. We assume that the origin is a degenerate singular point for these differential systems. We characterize when the singular point is of focus-centre type in a generic case. The problem of its local stability is also considered. We compute the first generalized Lyapunov constant when some non-degeneracy conditions are assumed.

DYNAMICS OF SOME RATIONAL DISCRETE DYNAMICAL SYSTEMS VIA INVARIANTS

We consider several discrete dynamical systems for which some invariants can be found. Our study includes complex Mobius transformations as well as the third-order Lyness recurrence.

LIMIT CYCLES FOR GENERALIZED ABEL EQUATIONS

This paper deals with the problem of finding upper bounds on the number of periodic solutions of a class of one-dimensional nonautonomous differential equations: those with the right-hand sides being polynomials of degree n and whose coefficients are real smooth one-periodic functions. The case n = 3 gives the so-called Abel equations which have been thoroughly studied and are well understood. We consider two natural generalizations of Abel equations. Our results extend previous works of Lins Neto and Panov and try to step forward in the understanding of the case n > 3. They can be applied, as well, to control the number of limit cycles of some planar ordinary differential equations.

Global periodicity and complete integrability of discrete dynamical systems

Consider the discrete dynamical system generated by a map F. It is said that it is globally periodic if there exists a natural number p such that F p (x)=x for all x in the phase space. On the other hand, it is called completely integrable if it has as many functionally independent first integrals as the dimension of the phase space. In this paper, we relate both concepts. We also give a large list of globally periodic dynamical systems together with a complete set of their first integrals, emphasizing the ones coming from difference equations.

Global asymptotic stability of differential equations in the plane

The problem of determining the basin of attraction of equilibrium points is of paramount importance for applications of stability theory. Local conditions which guarantee the existence of small basins of attraction, such as tr L 0, where L is the linear part of the planar system at an equilibrium point, are well known. This paper is concerned with sufficient conditions which guarantee that the basin of attraction of an equilibrium point of a Q?’ planar system of differential equations x’ =f(x) is the whole x-space R*. In this context, the fundamental problem, yet unsolved, is the following: Consider an autonomous system of differential equations x’=f(x) (’ = d/dt),

On Periodic Rational Difference Equations of Order k

This paper is devoted to the study of which rational difference equations of order k, with non negative coefficients, are periodic. Our main result is that for and for only the well known periodic difference equations and their natural extensions appear.

Studying discrete dynamical systems through differential equations

Abstract In this paper we consider dynamical systems generated by a diffeomorphism F defined on U an open subset of R n , and give conditions over F which imply that their dynamics can be understood by studying the flow of an associated differential equation, x ˙ = X ( x ) , also defined on U . In particular the case where F has n − 1 functionally independent first integrals is considered. In this case X is constructed by imposing that it shares with F the same set of first integrals and that the functional equation μ ( F ( x ) ) = det ( D F ( x ) ) μ ( x ) , x ∈ U , has some non-zero solution, μ. Several examples for n = 2 , 3 are presented, most of them coming from several well-known difference equations.