An explicit expression for the first return map in the center problem

The classical center-focus problem posed by H. Poincaré in 1880's asks about the characterization of planar polynomial vector fields such that all their integral trajectories are closed curves whose interiors contain a fixed point, a center. In this paper, we present a method allowing for the first time to obtain an explicit expression for the first return map in the center problem.     

An algebraic model for the center problem

We construct an algebraic model for the Center Problem for equation . This problem is related to the classical Poincaré Center-Focus problem for polynomial vector fields.     

On the cyclicity of weight-homogeneous centers

Let W be a weight-homogeneous planar polynomial differential system with a center. We find an upper bound of the number of limit cycles which bifurcate from the period annulus of W under a generic polynomial perturbation. We apply this result to a particular family of planar polynomial systems having a nilpotent center without meromorphic first integral.     

Vanishing of higher-order moments on Lipschitz curves

We study some algebraic and topological objects that appear naturally in the study of the center problem for the ordinary differential equation . In particular, we give a topological characterization of Lipschitz curves defined by the first integrals of the coefficients of this equation such that all moments of order?n, n∈N, vanish on them.     

On the number of limit cycles of some systems on the cylinder

In this article we give two criteria for bounding the number of non-contractible limit cycles of a family of differential systems on the cylinder. This family includes Abel equations as well as the polar expression of several types of planar polynomial systems given by the sum of three homogeneous vector fields.     

Hamiltonians and conjugate Hamiltonians of some fourth-order nonlinear ODEs

We first derive the Lagrangians of the reduced fourth-order ordinary differential equations studied by Kudryashov under the assumption that they satisfy the conditions stated by Fels [M.E. Fels, The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations, Trans. Amer. Math. Soc. 348, 1996, 5007-5029], using Jacobi’s last multiplier technique. In addition we derive the Hamiltonians of these equations using the Jacobi-Ostrogradski theory. Next, we derive the conjugate Hamiltonian equations for such fourth-order equations passing the Painlevé test. Finally, we investigate the conjugate Hamiltonian formulation of certain additional equations belonging to this family.     

Control sets and their boundaries under parameter variation

Control sets, i.e., maximal sets of approximate controllability, are described for systems with parameter-dependent control range. If the so-called inner-pair condition is satisfied, it is known that generically they change continuously under parameter variation while mergers of control sets produce discontinuous transitions. The inner-pair condition is proved for a class of control-affine systems on . Furthermore, continuity results for exit and entrance boundaries of control sets are given both for one control set that changes continuously and for two merging control sets. The results are illustrated by means of the controlled escape equation.     

Branches of forced oscillations in degenerate systems of second-order ODEs

We use fixed point index methods to study the set of forced oscillations in periodically perturbed systems of ODEs on manifolds. We prove the existence of branches of periodic solutions for a particular class of system where, contrary to the usual ‘nondegeneracy’ assumption, the leading vector field is neither trivial nor has a set of compact zeros.     

Simultaneous bifurcation of limit cycles from a linear center with extra singular points

The period annuli of the planar vector field x^′=−yF(x,y)$x^{\prime }=-yF(x,y)$, y^′=xF(x,y)$y^{\prime }=xF(x,y)$, where the set {F(x,y)=0}$\{F(x,y)=0\}$ consists of k   different isolated points, is defined by k+1$k+1$ concentric annuli. In this paper we perturb it with polynomials of degree n and we study how many limit cycles bifurcate, up to a first order analysis, from all the period annuli simultaneously in terms of k and n  . Additionally, we prove that the associated Abelian integral is piecewise rational and, when k=1$k=1$, the provided upper bound is reached. Finally, the case k=2$k=2$ is also treated.     

On a closed orbit of some constrained system

We study on what one calls a constrained system of ODEs on It consists of two ordinary differential equations and an algebraic equation with respect to three unknown functions. We seek closed orbits of such a system. A necessary and sufficient condition for the system to have non-trivial closed orbits is given. Elementary tools such as Lyapunov functions and Poincaré’s index theory are used in the proof of the result. As an application we consider a constrained system associated with a non-constraint system of ODEs called the modified Bonhöffer-van der Pol system.     

A system of ODEs with a hysteresis effect

The paper deals with a class of ordinary differential systems which contains a differential inclusion describing input–output relations of hysteresis type. Existence and uniqueness of local and global solutions of the systems under consideration are proved.     

Dynamics of a class of ODEs more general than almost periodic

A temporally global solution, if it exists, of a nonautonomous ordinary differential equation need not be periodic, almost periodic or almost automorphic when the forcing term is periodic, almost periodic or almost automorphic, respectively. An alternative class of functions extending periodic and almost periodic functions which has the property that a bounded temporally global solution solution of a nonautonomous ordinary differential equation belongs to this class when the forcing term does is introduced here. Specifically, the class of functions consists of uniformly continuous functions, defined on the real line and taking values in a Banach space, which have pre-compact ranges. Besides periodic and almost periodic functions, this class also includes many nonrecurrent functions. Assuming a hyperbolic structure for the unperturbed linear equation and certain properties for the linear and nonlinear parts, the existence of a special bounded entire solution, as well the existence of stable and unstable manifolds of this solution are established. Moreover, it is shown that this solution and these manifolds inherit the temporal behaviour of the vector field equation. In the stable case it is shown that this special solution is the pullback attractor of the system. A class of infinite dimensional examples involving a linear operator consisting of a time independent part which generates a C₀-semigroup plus a small time dependent part is presented and applied to systems of coupled heat and beam equations.     

On Hopf bifurcation in non-smooth planar systems

As we know, for non-smooth planar systems there are foci of three different types, called focus-focus (FF), focus-parabolic (FP) and parabolic-parabolic (PP) type respectively. The Poincaré map with its analytical property and the problem of Hopf bifurcation have been studied in Coll et al. (2001) [3] and Filippov (1988) [6] for general systems and in Zou et al. (2006) [13] for piecewise linear systems. In this paper we also study the problem of Hopf bifurcation for non-smooth planar systems, obtaining new results. More precisely, we prove that one or two limit cycles can be produced from an elementary focus of the least order (order 1 for foci of FF or FP type and order 2 for foci of PP type) (Theorem 2.3), different from the case of smooth systems. For piecewise linear systems we prove that 2 limit cycles can appear near a focus of either FF, FP or PP type (Theorem 3.3).     

Solutions of second-order and fourth-order ODEs on the half-line

We start by studying the existence of positive solutions for the differential equation

u^″=a(x)u−g(u),     

Melnikov theory for nonlinear implicit ODEs

We apply dynamical system methods and Melnikov theory to study small amplitude perturbation of some implicit differential equations. In particular we show persistence of such orbits connecting singularities in finite time provided a Melnikov like condition holds.     

On the critical points of the flight return time function of perturbed closed orbits

We deal here with planar analytic systems $\dot{x}=X(x,\epsilon )$ which are small perturbations of a period annulus. For each transversal section Σ to the unperturbed orbits we denote by $T^{\mathrm{\Sigma }}(q,\epsilon )$ the time needed by a perturbed orbit that starts from $q\in \mathrm{\Sigma }$ to return to Σ. We call this the flight return time function. We say that the closed orbit Γ of $\dot{x}=X(x,0)$ is a continuable critical orbit in a family of the form $\dot{x}=X(x,\epsilon )$ if, for any $q\in \mathrm{\Gamma }$ and any Σ that passes through q, there exists $q^{\epsilon }\in \mathrm{\Sigma }$ a critical point of $T^{\mathrm{\Sigma }}(?,\epsilon )$ such that $q^{\epsilon }\to q$ as $\epsilon \to 0$. In this work we study this new problem of continuability.In particular we prove that a simple critical periodic orbit of $\dot{x}=X(x,0)$ is a continuable critical orbit in any family of the form $\dot{x}=X(x,\epsilon )$. We also give sufficient conditions for the existence of a continuable critical orbit of an isochronous center $\dot{x}=X(x,0)$.     

Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields

We provide normal forms and the global phase portraits in the Poincaré disk for all Hamiltonian nilpotent centers of linear plus cubic homogeneous planar polynomial vector fields.     

Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields

We provide normal forms and the global phase portraits in the Poincaré disk for all the Hamiltonian linear type centers of linear plus cubic homogeneous planar polynomial vector fields.     

A cubic system with twelve small amplitude limit cycles

In this paper, the bifurcation of limit cycles for a cubic polynomial system is investigated. By the computation of the singular point values, we prove that the system has 12 small amplitude limit cycles. The process of the proof is algebraic and symbolic.