Related papers: Uniqueness of Limit Cycles for Quadratic Vector Fi…

In the paper, we first give the least upper bound formula on the number of centers of planar real polynomial Hamiltonian vector fields. This formula reveals that the greater the number of invariant straight lines of the vector field and the…

Let $x'=S(t,x)$ be a differential equation in the cylinder, linear piecewise in $x$ and with trigonometric coefficients in $t$. In this paper, we provide an upper bound on the number of limit cycles in terms of the number of regions of the…

In this paper, we will interest in finding the number of zeros of the quadratic forms over finite fields. We will apply the tool for finding the number of rational points of supersingular curves in [6]. We will give some more tools for…

Let $q$ be a non-degenerate quadratic form defined on an $F$ vector space $V$ and $a \in F$. We consider the Cayley graph on $V$ with generating set $\{x \in V \mid q(x) = a\}$ and study its diameter and girth. In particular, if $F$ is a…

Cyclic motions in vertebrates, including heart beating, breathing and walking, are derived by a network of biological oscillators having fascinating features such as entrainment, environment adaptation, and robustness. These features…

In this paper we are concerned with determining lower bounds of the number of limit cycles for piecewise polynomial holomorphic systems with a straight line of discontinuity. We approach this problem with different points of view: study of…

We study the iterated limit of a quaternary of means of four terms through the period map from the family of cyclic fourfold coverings of the complex projective line branching at six points to the three-dimensional complex ball…

We estimate the expected number of limit cycles situated in a neighbourhood of the origin of a planar polynomial vector field. Our main tool is a distributional inequality for the number of zeros of some families of univariate holomorphic…

Let $X$ be a projective variety over a number field $K$ endowed with a height function associated to an ample line bundle on $X$. Given an algebraic extension $F$ of $K$ with a sufficiently big Northcott number, we can show that there are…

In this paper, we study the analytical property of the Poincare return map and the generalized focal values of an analytical planar system with a nilpotent focus or center. Then we use the focal values and the map to study the number of…

A Hamiltonian cycle of a graph is a closed path which visits each of the vertices once and only once. In this article, Hamiltonian cycles on planar random lattices are considered. The generating function for the number of Hamiltonian cycles…

In recent papers we have introduced a method for the study of limit cycles of the Lienard system: dot{x}=y-F(x), dot{y}=-x, where F(x) is an odd polynomial. The method gives a sequence of polynomials R_n(x), whose roots are related to the…

We give an effective method for controlling the maximum number of limit cycles of some planar polynomial systems. It is based on a suitable choice of a Dulac function and the application of the well-known Bendixson-Dulac Criterion for…

Motivated by the controllability/reachability problems for switched linear control systems and some classes of nonlinear (mechanical) control systems we address a related problem of existence of a cyclic vector for an associative (matrix)…

We provide an upper bound for the number of limit cycles that planar polynomial differential systems of a given degree may have. The bound turns out to be a polynomial of degree four in the degree of the system. The strategy brings together…

Li\'enard equations of the form $\ddot{x}+\epsilon f(x)\dot{x}+x=0$, with $f(x)$ an even function, are considered in the weakly nonlinear regime ($\epsilon\to 0$). A perturbative algorithm for obtaining the number, amplitude and shape of…

A. Gasull shared a list of 33 open problems in low dimensional dynamical systems in his work in 2021. The second part of Problem 3 is about whether the limit cycle of a quasi-homogeneous system $ \dot{x}=y,\; \dot{y}=-x^3+\alpha x^2y+y^3 $…

By introducing a max-plus dynamical system having limit cycles, we discuss their periodicity, especially the number of discrete states in them. We also find that quasi-periodic cycles exist depending on the bifurcation parameter in the…

We will consider two special families of polynomial perturbations of the linear center. For the resulting perturbed systems, which are generalized Li\'enard systems, we provide the exact upper bound for the number of limit cycles that…

The geometry of algebraic curves over finite fields is a rich area of research. In previous work, the authors investigated a particular aspect of the geometry over finite fields of the classical unit circle, namely how the number of…