Tangential Hilbert problem for perturbations of hyperelliptic Hamiltonian systems.
The analysis of epidemic network model with infectious force in latent and infected period.
The analytic Carathéodory conjecture.
The existence of limit cycle for perturbed bilinear systems
In this paper, the feedback control for a class of bilinear control systems with a small parameter is proposed to guarantee the existence of limit cycle. We use the perturbation method of seeking in approximate solution as a finite Taylor expansion of the exact solution. This perturbation method is to exploit the “smallness” of the perturbation parameter to construct an approximate periodic solution. Furthermore, some simulation results are given to illustrate the existence of a limit cycle for...
The first derivative of period function of a plane vector field.
The algorithm of the successive derivatives introduced in [5] was implemented in [7], [8]. This algorithm is based on the existence of a decomposition of 1-forms associated to the relative cohomology of the Hamiltonian function which is perturbed. We explain here how the first step of this algorithm gives also the first derivative of the period function. This includes, for instance, new presentations of formulas obtained by Carmen Chicone and Marc Jacobs in [3].
The General Differential Operators Generated by a Quasi-Differential Expressions with their Interior Singular Points
The general ordinary quasi-differential expression M of n-th order with complex coefficients and its formal adjoint M + are considered over a regoin (a, b) on the real line, −∞ ≤ a < b ≤ ∞, on which the operator may have a finite number of singular points. By considering M over various subintervals on which singularities occur only at the ends, restrictions of the maximal operator generated by M in L2|w (a, b) which are regularly solvable with respect to the minimal operators T0 (M ) and T0...
The nonlinear limit-point/limit-circle problem for higher order equations
We describe the nonlinear limit-point/limit-circle problem for the -th order differential equation The results are then applied to higher order linear and nonlinear equations. A discussion of fourth order equations is included, and some directions for further research are indicated.
The null divergence factor.
Let (P,Q) be a C1 vector field defined in a open subset U ⊂ R2. We call a null divergence factor a C1 solution V (x, y) of the equation P ∂V/∂x + Q ∂V/ ∂y = ( ∂P/∂x + ∂Q/∂y ) V. In previous works it has been shown that this function plays a fundamental role in the problem of the center and in the determination of the limit cycles. In this paper we show how to construct systems with a given null divergence factor. The method presented in this paper is a generalization of the classical Darboux method...
The number of limit cycles of a quintic Hamiltonian system with perturbation.
The period function near a polycycle with two semi-hyperbolic vertices
The period of a whirling pendulum
The period function of a planar parameter-depending Hamiltonian system is examined. It is proved that, depending on the value of the parameter, it is either monotone or has exactly one critical point.
The rate of satiation in a cooperative system
The second term of the asymptotics of the monodromy map in case of two even edges of Newton diagram.
The XVI-th Hilbert problem about limit cycles
1. Introduction. The XVI-th Hilbert problem consists of two parts. The first part concerns the real algebraic geometry and asks about the topological properties of real algebraic curves and surfaces. The second part deals with polynomial planar vector fields and asks for the number and position of limit cycles. The progress in the solution of the first part of the problem is significant. The classification of algebraic curves in the projective plane was solved for degrees less than 8. Among general...
Théorèmes de finitude pour les variétés pfaffiennes
On introduit, dans ce travail, une hypothèse sur le spiralement d’une feuille d’un feuilletage analytique réel de codimension un (hypersurface pfaffienne). On en tire des résultats très généraux de finitude du type de Khovanskii. Des exemples précis montrent la généralité de ces hypersurfaces pfaffiennes. Une description complété des bouts de telles variétés en dimension trois est donnée.
Three-dimensional cooperative irreducible systems with a first integral
Topologická klasifikácia diferenciálnych rovníc a štrukturálna stabilita
Totally bounded differential polynomial systems in
Totally bounded differential systems in are defined as having all trajectories bounded. By Dulac’s finiteness theorem it is proved that totally bounded polynomial systems exhibit an unbounded «annulus» of cycles. The portrait of the remaining trajectories is examined in the case the system has, in , a unique singular point. Work is in progress concerning the study of totally bounded polynomial systems with two singular points.
Transformation groups on real plane and their differential invariants.
Transformation to Liénard form.