Planar Polynomial Vector Fields Research Articles

On a variant of Hilbert’s 16th problem

Abstract We study the number of limit cycles that a planar polynomial vector field can have as a function of its number m of monomials. We prove that the number of limit cycles increases at least quadratically with m and we provide good lower bounds for m ⩽ 10 .

On the structural instability of non-hyperbolic limit cycles on planar polynomial vector fields

On the structural instability of non-hyperbolic limit cycles on planar polynomial vector fields

Global planar dynamics with a star node and contracting nonlinearity

This is a complete study of the dynamics of polynomial planar vector fields whose linear part is a multiple of the identity and whose nonlinear part is a contracting homogeneous polynomial. The contracting nonlinearity provides the existence of an invariant circle and allows us to obtain a classification through a complete invariant for the dynamics, extending previous work by other authors that was mainly concerned with the existence and number of limit cycles. The general results are also applied to some classes of examples: definite nonlinearities, Z2⊕Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf {Z}}_2\\oplus {\ extbf {Z}}_2$$\\end{document} symmetric systems and nonlinearities of degree 3, for which we provide complete sets of phase-portraits.

Global dynamics of a class of planar semi-homogeneous polynomial vector fields with degree $$2-3$$

Global dynamics of a class of planar semi-homogeneous polynomial vector fields with degree $$2-3$$

Counting configurations of limit cycles and centers

We present several results on the determination of the number and distribution of limit cycles or centers for planar systems of differential equations. In most cases, the study of a recurrence is one of the key points of our approach. These results include the counting of the number of configurations of stabilities of nested limit cycles, the study of the number of different configurations of a given number of limit cycles, the proof of some quadratic lower bounds for Hilbert numbers and some questions about the number of centers for planar polynomial vector fields.

The improved Euler-Jacobi formula and the planar polynomial vector fields of degree 2-4

The new Euler-Jacobi formula for double points provides an algebraic relation between the singular points of a polynomial vector field and their topological indices. Using this formula we obtain the configuration of the singular points together with their topological indices for the planar quadratic--quartic polynomial differential systems when these systems have seven finite singular points being one of them with multiplicity two. The case with eight finite singular points has already been solved using the classical Euler-Jacobi formula.

Algebraic Integrability of Planar Polynomial Vector Fields by Extension to Hirzebruch Surfaces

We study algebraic integrability of complex planar polynomial vector fields \(X=A (x,y)(\partial /\partial x) + B(x,y) (\partial /\partial y) \) through extensions to Hirzebruch surfaces. Using these extensions, each vector field X determines two infinite families of planar vector fields that depend on a natural parameter which, when X has a rational first integral, satisfy strong properties about the dicriticity of the points at the line \(x=0\) and of the origin. As a consequence, we obtain new necessary conditions for algebraic integrability of planar vector fields and, if X has a rational first integral, we provide a region in \({\mathbb {R}}_{\ge 0}^2\) that contains all the pairs (i, j) corresponding to monomials \(x^i y^j\) involved in the generic invariant curve of X.

On the Hilbert number for piecewise linear vector fields with algebraic discontinuity set

The second part of Hilbert’s sixteenth problem consists in determining the upper bound H(n) for the number of limit cycles that planar polynomial vector fields of degree n can have. For n≥2, it is still unknown whether H(n) is finite or not. The main achievements obtained so far establish lower bounds for H(n). Regarding asymptotic behavior, the best result says that H(n) grows as fast as n2log(n). Better lower bounds for small values of n are known in the research literature. In the recent paper “Some open problems in low dimensional dynamical systems”, by A. Gasull, Problem 18 proposes a different Hilbert’s sixteenth type problem, namely improving the lower bounds for L(n), n∈N, which is defined as the maximum number of limit cycles that planar piecewise linear differential systems with two zones separated by a branch of an algebraic curve of degree n can have. So far, L(n)≥[n/2],n∈N, is the best known general lower bound. Again, better lower bounds for small values of n are known in the research literature. Here, by using a recently developed second order Melnikov method for nonsmooth systems with nonlinear discontinuity manifold, it is shown that L(n) grows as fast as n2. This will be achieved by providing lower bounds for L(n), which improve previous estimates for n≥4.

The improved Euler–Jacobi formula and the planar cubic polynomial vector fields in ℝ2

The new Euler–Jacobi formula for points with multiplicity two provides an algebraic relation between the singular points of a polynomial vector field and their topological indices. Using this formula we obtain the configuration of the singular points together with their topological indices for the planar cubic polynomial differential systems when these systems have eight finite singular points, being one of them with multiplicity two. The case with nine finite singular points has already been solved using the classical Euler–Jacobi formula.

An efficient method for computing Liouvillian first integrals of planar polynomial vector fields

Here we present an efficient method to compute Darboux polynomials for polynomial vector fields in the plane. This approach is restricted to polynomial vector fields presenting a Liouvillian first integral (or, equivalently, to rational first order differential equations (rational 1ODEs) presenting a Liouvillian general solution). The key to obtaining this method was to separate the procedure of solving the (non-linear) algebraic system resulting from the equation that translates the condition for the existence of a Darboux polynomial (i.e., from the equation D(p)=qp) into feasible steps (procedures that require less memory consumption). We also present a brief performance analysis of the algorithms developed.

Orbitally symmetric systems with applications to planar centers

<p style='text-indent:20px;'>We present a generalization of the most usual symmetries in differential equations known as the time-reversibility and the equivariance ones. We check that the typical properties are also valid for the new definition that unifies both. With it, we are able to present new families of planar polynomial vector fields having equilibrium points of center type. Moreover, we provide the highest lower bound for the local cyclicity of an equilibrium point of polynomial vector fields of degree 6, <inline-formula><tex-math id="M1">\begin{document}$ M(6)\ge 48. $\end{document}</tex-math></inline-formula></p>

Local cyclicity in low degree planar piecewise polynomial vector fields

In this work, we are interested in isolated crossing periodic orbits in planar piecewise polynomial vector fields defined in two zones separated by a straight line. In particular, in the number of limit cycles of small amplitude. They are all nested and surrounding one equilibrium point or a sliding segment. We provide lower bounds for the local cyclicity for planar piecewise polynomial systems, Mpc(n), with degrees 2, 3, 4, and 5. More concretely, Mpc(2)≥13,Mpc(3)≥26,Mpc(4)≥40, and Mpc(5)≥58. The computations use parallelization algorithms.

Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles

<p style='text-indent:20px;'>We provide normal forms and the global phase portraits in the Poincaré disk for all Hamiltonian planar polynomial vector fields of degree 3 symmetric with respect to the <inline-formula><tex-math id="M1">\begin{document}$ x- $\end{document}</tex-math></inline-formula>axis having a nilpotent saddle at the origin.

Nilpotent saddles of linear plus cubic homogeneous polynomial reversible vector fields

We provide normal forms and the global phase portraits in the Poincaré disk for all planar polynomial vector fields of the form lineal plus cubic homogeneous that are symmetric with respect to the x-axis or to the y-axis and having a nilpotent saddle at the origin.

Nonintegrability of the unfoldings of codimension-two bifurcations

Codimension-two bifurcations are fundamental and interesting phenomena in dynamical systems. Fold-Hopf and double-Hopf bifurcations are the most important among them. We study the unfoldings of these two codimension-two bifurcations, and obtain sufficient conditions for their nonintegrability in the meaning of Bogoyavlenskij. We reduce the problems of the unfoldings to those of planar polynomial vector fields and analyze the nonintegrability of the planar vector fields, based on Ayoul and Zung’s version of the Morales–Ramis theory. New useful criteria for nonintegrability of planar polynomial vector fields are also obtained. The approaches used here are applicable to many problems including circular symmetric systems.

A Note on the Lyapunov and Period Constants

It is well known that the number of small amplitude limit cycles that can bifurcate from the origin of a weak focus or a non degenerated center for a family of planar polynomial vector fields is governed by the structure of the so called Lyapunov constants, that are polynomials in the parameters of the system. These constants are essentially the coefficients of the odd terms of the Taylor development at zero of the displacement map. Although many authors use that the coefficients of the even terms of this map belong to the ideal generated by the previous odd terms, we have not found a proof in the literature. In this paper we present a simple proof of this fact based on a general property of the composition of one-dimensional analytic reversing orientation diffeomorphisms with themselves. We also prove similar results for the period constants. These facts, together with some classical tools like the Weirstrass preparation theorem, or the theory of extended Chebyshev systems, are used to revisit some classical results on cyclicity and criticality for polynomial families of planar differential equations.

Dynamical and Algebraic Analysis of Planar Polynomial Vector Fields Linked to Orthogonal Polynomials

In the present work, our goal is to establish a study of some families of quadratic polynomial vector fields connected to orthogonal polynomials that relate, via two different points of view, the qualitative and the algebraic ones. We extend those results that contain some details related to differential Galois theory as well as the inclusion of Darboux theory of integrability and the qualitative theory of dynamical systems. We conclude this study with the construction of differential Galois groups, the calculation of Darboux first integral, and the construction of the global phase portraits.

On the computation of Darboux first integrals of a class of planar polynomial vector fields

We study the class of planar polynomial vector fields admitting Darboux first integrals of the type ∏i=1rfiαi, where the αi's are positive real numbers and the fi's are polynomials defining curves with only one place at infinity. We show that these vector fields have an extended reduction procedure and give an algorithm which, from a part of the extended reduction of the vector field, computes a Darboux first integral for generic exponents.

On planar polynomial vector fields with elementary first integrals

We show that under rather general conditions a polynomial differential system having an elementary first integral already must admit a Darboux first integral, and we explicitly characterize the vector fields in this class. We also investigate some exceptional cases, i.e. equations admitting an elementary first integral but not a Darboux first integral. In particular we provide a rather detailed discussion of exceptional elementary first integrals built from algebraic functions of prime degree.

Bifurcations of limit cycles in complex networks of Hamiltonian systems and Lloyd's conjecture

This paper is motivated by Hilbert's sixteenth problem, concerning the number and distribution of limit cycles in planar polynomial vector fields. We propose an original method, inspired by the complex systems framework, in order to construct a family of Hamiltonian systems admitting non-degenerate centers located on the vertices of a planar network. We introduce two novel polynomial perturbations adapted to those systems, with the aim to produce a high number of limit cycles; the first one is oriented along the gradient of the unperturbed system, and the second one is again constructed with a complex systems approach. Using the Melnikov integral, we reach, in a small number of particular cases, the lower bound of cubical order for the Hilbert number H(n), conjectured by Lloyd in 1988.