Polynomial Vector Fields Of Degree Research Articles

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.

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.

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>

Lower bounds for the local cyclicity of centers using high order developments and parallelization

We are interested in small-amplitude isolated periodic orbits, so-called limit cycles, surrounding only one equilibrium point, that we locate at the origin. We develop a parallelization technique to study higher order developments, with respect to the parameters, of the return map near the origin. This technique is useful to study lower bounds for the local cyclicity of centers. We denote by M(n) the maximum number of limit cycles bifurcating from the origin via a degenerate Hopf bifurcation for a polynomial vector field of degree n. We get lower bounds for the local cyclicity of some known cubic centers and we prove that M(4)≥20, M(5)≥33, M(7)≥61, M(8)≥76, and M(9)≥88.

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.

The limit cycles of a class of quintic polynomial vector fields

Using the inverse integrating factor we study the limit cycles of a class of polynomial vector fields of degree $5$.

New lower bounds for the Hilbert numbers using reversible centers

In this paper we provide the best lower bounds, that are known up to now, for the Hilbert numbers of polynomial vector fields of degree N, , for small values of N. These limit cycles appear bifurcating from symmetric Darboux reversible centers with very high simultaneous cyclicity. The considered systems have, at least, three centers, one on the reversibility straight line and two symmetric outside it. More concretely, the limit cycles are in a three nests configuration and the total number of limit cycles is at least 2n + m, for some values of n and m. The new lower bounds are obtained using simultaneous degenerate Hopf bifurcations. In particular, and

A globally asymptotically stable polynomial vector field with rational coefficients and no local polynomial Lyapunov function

We give an explicit example of a two-dimensional polynomial vector field of degree seven that has rational coefficients, is globally asymptotically stable, but does not admit an analytic Lyapunov function even locally.

Polynomial Hamiltonian systems of degree 3 with symmetric nilpotent centers

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 x-axis having a nilpotent center at the origin.

Normal forms and global phase portraits of quadratic and cubic integrable vector fields having two nonconcentric circles as invariant algebraic curves

ABSTRACTIn this paper, we give the normal form of all planar polynomial vector fields of degree d ≤ 3 having two nonconcentric circles C1 and C2 as invariant algebraic curves and the function , with α and β real values, as first integral. Moreover, we classify all global phase portraits on the Poincaré disc of a subclass of these vector fields.

Limit cycles of discontinuous piecewise polynomial vector fields

When the first average function is non-zero we provide an upper bound for the maximum number of limit cycles bifurcating from the periodic solutions of the center x˙=−y((x2+y2)/2)m and y˙=x((x2+y2)/2)m with m≥1, when we perturb it inside a class of discontinuous piecewise polynomial vector fields of degree n with k pieces. The positive integers m, n and k are arbitrary. The main tool used for proving our results is the averaging theory for discontinuous piecewise vector fields.

The 16th Hilbert problem restricted to circular algebraic limit cycles

We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S admits at most S−1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S−1 algebraic limit cycles given by circles, and this number is reached.

Calculation of singular point quantities at infinity for a type of polynomial differential systems

The center problem at infinity is far to be solved in general. In this paper we develop a procedure to resolve it for a particular type of polynomial differential systems. The problem is solved by writing its concomitant differential equation in the complex coordinates introduced by Yirong Liu and by developing a new method of computation of the so called singular point quantities. This method is based on the transformation of infinity into the elementary origin. Finally, the investigation of center problem for the infinity of a particular family of planar polynomial vector fields of degree 5 is carried out to illustrate the main theoretical results. These involve extensive use of a Computer Algebra System, we have chosen to use Mathematica®.

On the polynomial vector fields on

Let X be a polynomial vector field of degree n on M, M = ℝm. The dynamics and the algebraic-geometric properties of the vector fields X have been studied intensively, mainly for the case when M = ℝm, and especially when n = 2. Several papers have been dedicated to the study of the homogeneous polynomial vector field of degree n on $\mathbb{S}^2$, mainly for the case where n = 2 and M = $\mathbb{S}^2$. But there are very few results on the non-homogeneous polynomial vector fields of degree n on $\mathbb{S}^2$. This paper attempts to rectify this slightly.

The 16th Hilbert problem on algebraic limit cycles

For real planar polynomial differential systems there appeared a simple version of the 16th Hilbert problem on algebraic limit cycles: Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree m? In [J. Llibre, R. Ramírez, N. Sadovskaia, On the 16th Hilbert problem for algebraic limit cycles, J. Differential Equations 248 (2010) 1401–1409] Llibre, Ramírez and Sadovskaia solved the problem, providing an exact upper bound, in the case of invariant algebraic curves generic for the vector fields, and they posed the following conjecture: Is 1 + ( m − 1 ) ( m − 2 ) / 2 the maximal number of algebraic limit cycles that a polynomial vector field of degree m can have? In this paper we will prove this conjecture for planar polynomial vector fields having only nodal invariant algebraic curves. This result includes the Llibre et al.ʼs as a special one. For the polynomial vector fields having only non-dicritical invariant algebraic curves we answer the simple version of the 16th Hilbert problem.

Limit cycles bifurcating from a perturbed quartic center

We consider the quartic center x ˙ = - yf ( x , y ) , y ˙ = xf ( x , y ) , with f( x, y) = ( x + a) ( y + b) ( x + c) and abc ≠ 0. Here we study the maximum number σ of limit cycles which can bifurcate from the periodic orbits of this quartic center when we perturb it inside the class of polynomial vector fields of degree n, using the averaging theory of first order. We prove that 4[( n − 1)/2] + 4 ⩽ σ ⩽ 5[( n − 1)/2] + 14, where [ η] denotes the integer part function of η.

Bifurcation of Limit Cycles from a Polynomial Degenerate Center

Abstract Using Melnikov functions at any order, we provide upper bounds for the maximum number of limit cycles bifurcating from the period annulus of the degenerate center ẋ = −y((x2 + y2)/2)m and ẏ = x((x2 + y2)/2)m with m ≥ 1, when we perturb it inside the whole class of polynomial vector fields of degree n. The positive integers m and n are arbitrary. As far as we know there is only one paper that provide a similar result working with Melnikov functions at any order and perturbing the linear center ẋ = −y, ẏ = x.

On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves

We give an upper bound for the maximum number N of algebraic limit cycles that a planar polynomial vector field of degree n can exhibit if the vector field has exactly k nonsingular irreducible invariant algebraic curves. Additionally we provide sufficient conditions in order that all the algebraic limit cycles are hyperbolic. We also provide lower bounds for N.

New Results on the Study of Z q -Equivariant Planar Polynomial Vector Fields

In this paper, we introduce some new results on the study of Zq-equivariant planar polynomial vector fields. The main conclusions are as follows. (1) For the Z2-equivariant planar cubic systems having two elementary foci, the characterization of a bi-center problem and shortened expressions of the first six Lyapunov constants are completely solved. The necessary and sufficient conditions for the existence of the bi-center are obtained. All possible first integrals are given. Under small Z2-equivariant cubic perturbations, the conclusion that there exist at most 12 small-amplitude limit cycles with the scheme \({\langle 6\amalg 6\rangle}\) is proved. (2) On the basis of mentioned work in (1), by considering the bifurcation of a global limit cycle from infinity, we show that under small Z2-equivariant cubic perturbations, such bi-center cubic system has at least 13 limit cycles with the scheme \({\langle 1 \langle 6\amalg 6\rangle\rangle}\), i.e., we obtain that the Hilbert number H(3) ≥ 13. (3) For the Z5-equivariant planar polynomial vector field of degree 5, we shown that such system has at least five symmetric centers if and only if it is a Hamltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z5-equivariant perturbations, the conclusion that perturbed system has at least 25 limit cycles with the scheme \({\langle 5\amalg 5\amalg 5\amalg 5\amalg 5\rangle}\) is rigorously proved. (4) For the Z6-equivariant planar polynomial vector field of degree 5, we proved that such system has at least six symmetric centers if and only if it is a Hamltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z6-equivariant perturbations, the conclusion that perturbed system has at least 24 limit cycles with the scheme \({\langle 4\amalg 4\amalg 4\amalg 4\amalg 4\amalg4\rangle}\) is rigorously proved. Two schemes of distributions of limit cycles are given.

Bifurcations of limit cycles in a -equivariant planar polynomial vector field of degree 7

The present paper intends to explore the number and distribution of limit cycles for a class of perturbed Z4-equivariant planar Hamiltonian system of degree 7. The bifurcation theory of planar dynamical systems and the method of detection functions are applied to study the given perturbed system. By controlling the perturbation parameters, an example of a special perturbed Z4-equivariant vector field having 41 limit cycles is found. The distribution of the above 41 limit cycles is also given. The results are significant to the study of weaken Hilbert's 16th problem and to the acquaintance of the characteristics of higher-order equivariant vector fields.