Related papers: Isochronicity conditions for some planar polynomia…

We revisit the characterization of \emph{trivial} isochronous centers for planar polynomial Hamiltonian systems in degrees $5$ and $7$ obtained by Braun--Llibre--Mereu, and we formalize two conclusions suggested by their method. First, a…

Sufficient conditions for synchronization of coupled Lienard-type oscillators are investigated via averaging technique. Coupling considered here is pairwise, unidirectional, and described by a nonlinear function (whose graph resides in the…

We give a short proof of Urabe's criteria for the isochronicity of periodical solutions of the equation $\ddot{x}+g(x)=0$. We show that apart from the harmonic oscillator there exists a large family of isochronous potentials which must all…

Abel equations of the form $x'(t)=f(t)x^3(t)+g(t)x^2(t)$, $t \in [-a,a]$, where $a>0$ is a constant, $f$ and $g$ are continuous functions, are of interest because of their close relation to planar vector fields. If $f$ and $g$ are odd…

S-systems are simple examples of power-law dynamical systems (polynomial systems with real exponents). For planar S-systems, we study global stability of the unique positive equilibrium and solve the center problem. Further, we construct a…

This paper investigates the monotonicity of the period function associated with planar Hamiltonian systems of the form $H(x,y) = F(x) + G(y)$. We establish sufficient conditions ensuring the monotonicity of the period function corresponding…

We study the center-focus problem for planar polynomial vector fields, which can be viewed as a local version of Hilbert's 16th problem. Based on a Lyapunov function approach, we establish novel results regarding the center-focus…

In this paper we determine the centers of quasi-homogeneous polynomial planar vector fields of degree 0, 1, 2, 3 and 4. In addition, in every case we make a study of the reversibility and the analytical integrability of each one of the…

In this paper, we study the topological properties of complex polynomial Hamiltonian differential systems of degree $n$ having an isochronous center. Firstly, we prove that if the critical level curve possessing an isochronous center…

In this paper, we study the many-to-many matching problem on planar point sets with integer coordinates: Given two disjoint sets $R,B \subset [\Delta]^2$ with $|R|+|B|=n$, the goal is to select a set of edges between $R$ and $B$ so that…

In this paper, we construct two classes of planar polynomial Hamiltonian systems having a center at the origin, and obtain the lower bounds for the number of critical periods for these systems. For polynomial potential systems of degree…

An oscillator is called isochronous if all motions have a common period. When the system is forced by a time-dependent perturbation with the same period the dynamics may change and the phenomenon of resonance can appear. In this context,…

For the planar $N$-centre problem $$ \ddot x = - \sum_{i=1}^N \frac{m_i (x-c_i)}{| x - c_i|^{\alpha+2}}, \qquad x \in \mathbb{R}^2 \setminus \{ c_1,\ldots,c_N \}, $$ where $m_i > 0$ for $i=1,\ldots,N$ and $\alpha \in [1,2)$, we prove the…

We study general random dynamical systems of continuous maps on some compact metric space. Assuming a local contraction condition and uniqueness of the stationary measure, we establish probabilistic limit laws such as the central limit…

In this paper we study the family of planar hybrid differential systems formed by two linear centers and a polynomial reset map of any degree. We study their limit cycles and also provide examples of these hybrid systems exhibiting chaotic…

Let $P$ be a planar $n$-gon with the sidelengths $a_1, \ldots, a_n$ and let us denote by $L=L(P)$ the corresponding planar polygonal linkage. We are concerned with the problem of finding conditions on the sidelengths $a_i$ which guarantee…

In this paper, we study a Lienard system of the form dot{x}=y-F(x), dot{y}=-x, where F(x) is an odd polynomial. We introduce a method that gives a sequence of algebraic approximations to the equation of each limit cycle of the system. This…

Consider the planar linear switched system $\dot x(t)=u(t)Ax(t)+(1-u(t))Bx(t),$ where $A$ and $B$ are two $2\times2$ real matrices, $x \in \R^2$, and $u(.):[0,\infty[\to\{0,1\}$ is a measurable function. In this paper we consider the…

Consider a family of planar polynomial systems $\dot x = y^{2l-1} - x^{2k+1}, \dot y =-x +m y^{2s+1},$ where $l,k,s\in\mathbb{N^*},$ $2\le l \le 2s$ and $m\in\mathbb{R}.$ We study the center-focus problem on its origin which is a monodromic…

These last years an increasing interest appeared for studying the planar discontinuous piecewise differential systems motivated by the rich applications in modelling real phenomena. One of the difficulties for understanding the dynamics of…