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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…

We prove that homoclinic classes for a residual set of C^1 vector fields X on closed n-manifolds are maximal transitive and depend continuously on periodic orbit data. In addition, X does not exhibit cycles formed by homoclinic classes. We…

We extend the classical Fuller index, and use this to prove that for a certain general class of vector fields $X$ on a compact smooth manifold, if a homotopy of smooth non-singular vector fields starting at $X$ has no sky catastrophes as…

The object of this paper is to address the following question: When is a polynomial vector field on $\mathbb{C}^2$ completely determined (up to affine equivalence) by the spectra of its singularities? We will see that for quadratic vector…

Let X be a compact complex manifold whose anti-canonical line bundle is big. We show that X admits no non-trivial holomorphic vector fields if it is Gibbs stable (at any level). The proof is based on a vanishing result for measure…

In this paper we consider the complex vector spaces of holomorphic cross-sections of homogeneous holomorphic vector bundles over elliptic adjoint orbits, and provide a sufficient condition for the vector spaces to be finite dimensional in…

The quasi-homogeneous (and in general non-homogeneous) polynomial differential systems have been studied from many different points of view. In this paper, Center-focus determination and limit cycles bifurcation for $p:q$ homogeneous weight…

This article deals with the study of the number of limit cycles surrounding a critical point of a quadratic planar vector field, which, in normal form, can be written as $x'= a_1 x-y-a_3x^2+(2 a_2+a_5)xy + a_6 y^2$, $y'= x+a_1 y + a_2x^2+(2…

Smooth vector fields on $\mathbb{R}^n$ can be decomposed into the sum of a gradient vector field and divergence-free (solenoidal) vector field under suitable hypotheses. This is called the Helmholtz-Hodge decomposition (HHD), which has been…

We give lower bounds in terms of~$n,$ for the number of limit cycles of polynomial vector fields of degree~$n,$ having any prescribed invariant algebraic curve. By applying them when the ovals of this curve are also algebraic limit cycles…

We study the maximum number of limit cycles that can bifurcate from a degenerate center of a cubic homogeneous polynomial differential system. Using the averaging method of second order and perturbing inside the class of all cubic…

In this paper, by using Picard-Fuchs equations and Chebyshev criterion, we study the bifurcate of limit cycles for quadratic Hamilton system $S^{(2)}$ and $S^{(3)}$: $\dot{x}= y+2axy+by^2$, $\dot{y}=-x+x^2-ay^2$ with $a\in(-\frac{1}{2},1)$,…

We consider families of planar polynomial vector fields of degree $n$ and study the cyclicity of a type of unbounded polycycle~$\Gamma$ called hemicycle. Compactified to the Poincar\'e disc,~$\Gamma$ consists of an affine straight line…

It has been conjectured that for $N$ sufficiently large, there are no quadratic polynomials in $\bold Q[z]$ with rational periodic points of period $N$. Morton proved there were none with $N=4$, by showing that the genus~$2$ algebraic curve…

The goal of this article is to try understand where Hodge cycles on a singular complex projective variety X come from. As a first step we consider Hodge cycles on the maximal pure quotient $H^{2p}(X)/W_{2p-1}$, and introduce a class of…

The purpose of the present paper is to study the limit cycles of one-parameter perturbed plane Hamiltonian vector field $X_\varepsilon$ $$ X_\varepsilon : \left\{ \begin{array}{llr} \dot{x}=\;\; H_y+\varepsilon f(x,y)\\…

We determine several necessary and sufficient conditions for a closed almost-complex orbifold $Q$ with cyclic local groups to admit a nonvanishing vector field. These conditions are stated separately in terms of the orbifold Euler-Satake…

We describe some families of differentiable vector fields with the Hopf bifurcation at infinity, without assuming the continuous differentiability. These vector fields have isolated singular points on the plane, and the initial families are…

In this paper, applying a canonical system with field rotation parameters and using geometric properties of the spirals filling the interior and exterior domains of limit cycles, we solve first the problem on the maximum number of limit…

The second part of Hilbert's 16th problem concerns determining the maximum number $H(m)$ of limit cycles that a planar polynomial vector field of degree $m$ can exhibit. A natural extension to the three-dimensional space is to study the…