Anna Cima

ALGEBRAIC AND TOPOLOGICAL CLASSIFICATION IF THE HOMOGENEOUS CUBIC VECTOR FIELDS IN THE PLANE

Tsutomu Date and Masao Iri in [DI] gave an algebraic classification of systems I = P(x, y), I; = Q(.x, y), where P and Q are homogeneous polynomials of degree 2. For this, they used the classification of the binary cubic forms and also the simultaneous classification of a linear binary form and a cubic binary form given by the algebraic invariant theory. We begin by doing a similar study for systems .?? = P(x, y), ,’ = Q(x, y), where P and Q are homogeneous polynomials of degree three (i.e., cubic systems). The classification’s theorem of such systems is based on the classification of fourth-order binary forms. Gurevich in [Gu] did the classification of fourth-order binary forms on the field of complex numbers. Since we did not find the classification on the real domain, we adapt Gurevich’s proof to obtain it. In Section 1 we give some definitions and preliminary results, while in Section 2 we give the theorem of classification of fourth-order binary forms on the real domain. The method used in the proof (Caley’s method) let us obtain canonical forms of the fourth-order binary forms and the algebraic characteristics. Section 3 is devoted to obtaining the algebraic classification of systems .? = P(x, y), j = Q(x, y), where P and Q are homogeneous polynomials of degree 3. Given an arbitrary system X= (P, Q) with P and Q homogeneous polynomials of degree 3, we can know the equivalence-class at which it belongs through the algebraic characteristics. In Section 4 we study the phase-portraits of systems 1= P(x, y), j = Q(x, y), where P and Q are homogeneous polynomials of degree n and P and Q have no common factor. Such systems have been studied by J. Argemi in [A]. Here we give a shorter new proof of his results by using

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.

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.

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.

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.

Integrability and non-integrability of periodic non-autonomous Lyness recurrences

This paper studies non-autonomous Lyness-type recurrences of the form x n+2 = (a n + x n+1)/x n , where {a n } is a k-periodic sequence of positive numbers with primitive period k. We show that for the cases k ∈ {1, 2, 3, 6}, the behaviour of the sequence {x n } is simple (integrable), while for the remaining cases satisfying this behaviour can be much more complicated (chaotic). We also show that the cases where k is a multiple of 5 present some different features.

On 2- and 3-periodic Lyness difference equations

We describe the sequences given by the non-autonomous second-order Lyness difference equations , where is either a 2-periodic or a 3-periodic sequence of positive values and the initial conditions are also positive. We also show an interesting phenomenon of the discrete dynamical systems associated with some of these difference equations: the existence of one oscillation of their associated rotation number functions. This behaviour does not appear for the autonomous Lyness difference equations.

A polynomial class of Markus-Yamabe counterexamples

In the paper \cite{CEGHM} a polynomial counterexample to the Markus-Yamabe Conjecture and to the discrete Markus-Yamabe Question in dimension

NONAUTONOMOUS TWO-PERIODIC GUMOVSKI–MIRA DIFFERENCE EQUATIONS

n\ge 3