Search

In this paper we find an upper bound for the maximum number of limit cycles bifurcating from the periodic orbits of any planar polynomial quasi-homogeneous center, which can be obtained using first order averaging method. This result improves the upper bounds given in [7] . [ABSTRACT FROM AUTHOR]

This paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincaré–Bendixson regions by using transversal conics. We present several examples of known systems in the literature showing different features about limit cycles: hyperbolicity, Hopf bifurcation, sky-blue bifurcation, rotated vector fields, etc., for which the obtained Poincaré–Bendixson region allows to locate the limit cycles. Our method gives bounds for the bifurcation values of parametrical families of planar vector fields and intervals of existence of limit cycles. [ABSTRACT FROM AUTHOR]

We consider a planar autonomous real analytic differential system with a monodromic singular point p . We deal with the center problem for the singular point p . Our aim is to highlight some relations between the divergence of the system and the Poincaré–Liapunov constants of p when these are defined. [ABSTRACT FROM AUTHOR]

Recently some extensions of the classical Darboux integrability theory to autonomous and non-autonomous polynomial vector fields were completed. The classical Darboux integrability theory and its recent extensions are based on the existence of algebraic invariant hypersurfaces. However the algebraicity of the invariant hypersurfaces is not necessary and the unique necessary condition is the algebraicity of the cofactors associated to them. In this paper a more general extension of the classical Darboux integrability theory is established. [ABSTRACT FROM AUTHOR]

Abstract: We provide a recurrence formula for the coefficients of the powers of in the series expansion of the solutions around of the perturbed first-order differential equations. Using it, we give an averaging theory at any order in for the following two kinds of analytic differential equation: A planar polynomial differential system with a singular point at the origin can be transformed, using polar coordinates, to an equation of the previous form. Thus, we apply our results for studying the limit cycles of a planar polynomial differential systems. [Copyright &y& Elsevier]

In this paper, we characterize the universal centres of the ordinary differential equations [ABSTRACT FROM AUTHOR]

Abstract: In this work we study the existence and non-existence of finite invariant algebraic curves for a complex planar polynomial differential system having a Liouvillian first integral. [Copyright &y& Elsevier]

Abstract: The integrability problem consists of finding the class of functions a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit an elementary or Liouvillian first integral. We define Weierstrass integrability and we determine some Weierstrass integrable systems which are Liouvillian integrable. Inside this new class of integrable systems there are non-Liouvillian integrable systems. [Copyright &y& Elsevier]

Abstract: We consider linear homogeneous differential equations of the form where is a square matrix of , real and -periodic functions, with . We give several criteria on the matrix to prove the asymptotic stability of the trivial solution to equation . These criteria allow us to show that any finite configuration of cycles in can be realized as hyperbolic limit cycles of a polynomial vector field. [Copyright &y& Elsevier]

Abstract: In this note we give a family of planar polynomial differential systems with a prescribed hyperbolic limit cycle. This family constitutes a corrected and wider version of an example given in the work [M.A. Abdelkader, Relaxation oscillators with exact limit cycles, J. Math. Anal. Appl. 218 (1998) 308–312]. The result given in this note may be used to construct models of Liénard differential equations exhibiting a desired limit cycle. [Copyright &y& Elsevier]

Abstract: In this paper, we consider complex smooth and analytic vector fields in a neighborhood of a nondegenerate singular point. It is proved the equivalence between linearizability and commutation, i.e., the existence of a commuting vector field such that the Lie brackets . For complex smooth and analytic vector fields in the plane and in a neighborhood of a nondegenerate singular point, it is also proved the equivalence between integrability and the existence of a smooth vector field , such that is a normalizer of , i.e., . [Copyright &y& Elsevier]

In this paper we study algebraic solutions of the form h(x,y)=p1(x)y+p2(x), where p1(x) and p2(x) are real coprime polynomials and p1(x)≠0, for a planar real algebraic quadratic system. We analyze the existence of this kind of solutions depending on the coefficients of the system. [Copyright &y& Elsevier]

Abstract: In this Letter we study some aspects of the relationship between the existence of Darboux polynomials and additional polynomial first integrals in natural polynomial Hamiltonian systems with a finite number of degrees of freedom. More precisely, first we improve results of the paper of Maciejewski and Przybylska [A.J. Maciejewski, M. Przybylska, Phys. Lett. A 326 (2004) 219]; and after we answer two open questions presented in that paper. [ABSTRACT FROM AUTHOR]

Abstract We deal here with planar analytic systems x ˙ = X (x , ε) which are small perturbations of a period annulus. For each transversal section Σ to the unperturbed orbits we denote by T Σ (q , ε) the time needed by a perturbed orbit that starts from q ∈ Σ to return to Σ. We call this the flight return time function. We say that the closed orbit Γ of x ˙ = X (x , 0) is a continuable critical orbit in a family of the form x ˙ = X (x , ε) if, for any q ∈ Γ and any Σ that passes through q , there exists q ε ∈ Σ a critical point of T Σ (⋅ , ε) such that q ε → q as ε → 0. In this work we study this new problem of continuability. In particular we prove that a simple critical periodic orbit of x ˙ = X (x , 0) is a continuable critical orbit in any family of the form x ˙ = X (x , ε). We also give sufficient conditions for the existence of a continuable critical orbit of an isochronous center x ˙ = X (x , 0). Highlights • We define the notion of continuable critical orbit, which is a generalization of critical orbit. • We prove that a simple critical periodic orbit is a continuable critical orbit in any perturbed family. • We give sufficient conditions for the existence of a continuable critical orbit of an isochronous center. [ABSTRACT FROM AUTHOR]

In the present paper we tackle the problem of determining the multiplicity of periodic orbit as limit cycle of a planar differential system. We consider the particular case of a circumference as periodic orbit. We show that the conditions of multiplicity can be almost algebraically solvable. There are parameters in which these conditions depend transcendentally, as happens in the degenerate center-focus problem. Even though this difficulty, these transcendental dependence can be, in some sense, controlled because only a basis of fundamental functions appear. The appearance of this fundamental basis opens the path to approach these types of problems. We present several examples of families for which these conditions can be computed. [ABSTRACT FROM AUTHOR]

In this paper we study the maximum number of limit cycles that can bifurcate from a focus singular point p of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider p being a focus singular point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of p the differential system can always be brought, by means of a change to (generalized) polar coordinates ( r, θ), to an equation over a cylinder in which the singular point p corresponds to a limit cycle γ. This equation over the cylinder always has an inverse integrating factor which is smooth and non-flat in r in a neighborhood of γ. We define the notion of vanishing multiplicity of the inverse integrating factor over γ. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the singular point p in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse integrating factor in a neighborhood of many types of singular points, namely for the three types of focus considered in the previous paragraph and for any isolated singular point with at least one non-zero eigenvalue. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: This work deals with limit cycles of real planar analytic vector fields. It is well known that given any limit cycle Γ of an analytic vector field it always exists a real analytic function , defined in a neighborhood of Γ, and such that Γ is contained in its zero level set. In this work we introduce the notion of being an m-solution, which is a merely analytic concept. Our main result is that a limit cycle Γ is of multiplicity m if and only if is an m-solution of the vector field. We apply it to study in some examples the stability and the bifurcation of periodic orbits from some non-hyperbolic limit cycles. [Copyright &y& Elsevier]

Abstract: In this paper we give a very easy to compute necessary condition in the monodromy problem for all singular point of analytic differential systems in the real plane. Our main tool is considering the analytic function, angular speed, and studying its limit through straight lines to the singular point. [Copyright &y& Elsevier]