Semistability of motions and regular dependence of limit sets on points in general semi-sytems
Separatrices and singular points.
Simple examples of one-parameter planar bifurcations.
In this paper we give simple and low degree examples of one-parameter polynomial families of planar differential equations which present generic, codimension one, isolated, compact bifurcations. In contrast with some examples which appear in the usual text books each bifurcation occurs when the bifurcation parameter is zero. We study the total number of limit cycles that the examples present and we also make their phase portraits on the Poincaré sphere.
Singular points and Lie rotated vector fields.
Smooth linearization of centres
Smoothness of unordered curves in two-dimensional strongly competitive systems
It is known that in two-dimensional systems of ODEs of the form with (strongly competitive systems), boundaries of the basins of repulsion of equilibria consist of invariant Lipschitz curves, unordered with respect to the coordinatewise (partial) order. We prove that such curves are in fact of class .
Solution of the 1 : −2 resonant center problem in the quadratic case
The 1:-2 resonant center problem in the quadratic case is to find necessary and sufficient conditions (on the coefficients) for the existence of a local analytic first integral for the vector field . There are twenty cases of center. Their necessity was proved in [4] using factorization of polynomials with integer coefficients modulo prime numbers. Here we show that, in each of the twenty cases found in [4], there is an analytic first integral. We develop a new method of investigation of analytic...
Some properties of integral curves in a neighbourhood of planar singular points
Si studia l'andamento delle traiettorie di un sistema dinamico piano rappresentato dalle equazioni (1) del testo, nell'intorno di un punto singolare isolato.
Stabilité locale des systèmes quadratiques
Stable closed orbits in plane autonomous dynamical systems.
Studies on the differential equations of enzyme kinetics. I. Bimolecular scheme: simplified model.
Sufficient conditions for the existence of a center in polynomial systems of arbitrary degree.
In this paper, we consider polynomial systems of the form x' = y + P(x, y), y' = -x + Q(x, y), where P and Q are polynomials of degree n wihout linear part.For the case n = 3, we have found new sufficient conditions for a center at the origin, by proposing a first integral linear in certain coefficient of the system. The resulting first integral is in the general case of Darboux type.By induction, we have been able to generalize these results for polynomial systems of arbitrary degree.
Sur la croissance des intégrales des équations différentielles algébriques du premier ordre
Sur les points singuliers des équations différentielles
Sur les points singuliers des équations différentielles (suite)
Sur les solutions singulières des équations aux dérivées ordinaires du premier ordre
Sur une classe d'équations différentielles lineaires du deuxième ordre resolubles par quadratures
Sur une équation différentielle du premier ordre et du second degré
Surstabilité pour une équation différentielle analytique en dimension un
En rapport avec le problème du retard a la bifurcation, la notion de solution surstable est définie pour une famille d’équations différentielles analytiques avec un petit paramètre. Un théorème d’existence des solutions surstables est démontré pour des valeurs exceptionnelles d’un paramètre de contrôle. L’outil principal de la démonstration est un théorème de sommation qui constitue une généralisation d’un résultat de A. I. Neishtadt.