Period doubling bifurcations in a two-box model of the Brusselator
Two theorems about period doubling bifurcations are proved. A special case, where one multiplier of the homogeneous solution is equal to +1 is discussed in the Appendix.
Period function's convexity for Hamiltonian centers with separable variables
A convexity theorem for the period function T of Hamiltonian systems with separable variables is proved. We are interested in systems with non-monotone T. This result is applied to proving the uniqueness of critical orbits for second order ODE's.
Periodic solution of certain second order differential equation
Periodic solutions of a three-species food chain model.
Periodic solutions of the Rayleigh equation with damping of definite sign
The existence of a non-trivial periodic solution for the autonomous Rayleigh equation is proved, assuming conditions which do not imply that has a definite sign for large. A similar result is obtained for the periodically forced equation .
Perturbations of quadratic hamiltonian systems with symmetry
Phase portraits of planar vector fields: Computer proofs.
Polynomial Riccati equations with algebraic solutions
We consider the equations of the form dy/dx = y²-P(x) where P are polynomials. We characterize the possible algebraic solutions and the class of equations having such solutions. We present formulas for first integrals of rational Riccati equations with an algebraic solution. We also present a relation between the problem of algebraic solutions and the theory of random matrices.
Prolungamento dellacurva integrale del sistema di equazioni differenziali sull’insieme singolare. Teoria e applicazioni
Pseudo-abelian integrals on slow-fast Darboux systems
We study pseudo-abelian integrals associated with polynomial deformations of slow-fast Darboux integrable systems. Under some assumptions we prove local boundedness of the number of their zeros.
Quadratic Isochronous centers commute
We prove that every quadratic plane differential system having an isochronous center commutes with a polynomial differential system.
Quadratic systems equivalent by domains to a linear one: global phase portraits.
Quadratic systems with a unique finite rest point.
We study phase portraits of quadratic systems with a unique finite singularity. We prove that there are 111 different phase portraits without limit cycles and that 13 of them are realizable with exactly one limit cycle. In order to finish completely our study two problems remain open: the realization of one topologically possible phase portrait, and to determine the exact number of limit cycles for a subclass of these systems.
Quadratic systems with homoclinic cycles described by quintic curves.
Quadratic vector fields with a weak focus of third order.
We study phase portraits of quadratic vector fields with a weak focus of third order at the origin. We show numerically the existence of at least 20 different global phase portraits for such vector fields coming from exactly 16 different local phase portraits available for these vector fields. Among these 20 phase portraits, 17 have no limit cycles and three have at least one limit cycle.
Qualitative analysis for a class of plane systems.
Qualitative investigation of a two-dimensional Chua system.
Reductibilite d'un systeme differentiel lineaire (Reductibility of a Linear Differential System)
Relations between limit-point and Dirichlet properties of second-order difference operators.
Réversibilité et classification des centres nilpotents
Nous considérons un germe de 1-forme analytique dans dont le 1-jet est . Nous montrons que si l’équation définit un centre (i.e toutes les courbes solutions sont des cycles) il existe une involution analytique de préservant le portrait de phase du système. Géométriquement ceci signifie que les centres analytiques nilpotents sont obtenus par image réciproque par des applications pli. Un théorème de conjugaison équivariante permet d’obtenir une classification complète de ces centres.