Related papers: Isochronous Centers of Lienard Type Equations and …
Non-autonomous perturbations of isochronous systems in the plane are considered. It is assumed that the intensity of perturbations decays with time, and the frequency is asymptotically constant with the limiting value satisfying a resonance…
In this paper we are concerned with the boundedness of all solutions for the forced isochronous oscillator $$x''+V'(x)+g(x)=f(t),$$ where $V$ is a so-called $T$-isochronous potential, the perturbation $g$ is assumed to be bounded, and the…
We study a specific family of uniformly isochronous polynomial systems. Our results permit to solve a problem about centers of such systems.
In this paper, we study the integrability and linearization of a class of quadratic quasi-analytic switching systems. We improve an existing method to compute the focus values and periodic constants of quasi-analytic switching systems. In…
Isochronous waveform solutions of homogeneous Li\'enard equations are obtained by a modification of the nonlinear factorization method of Rosu and Cornejo-P\'erez. The scheme is based on the assumption that the intermediate function $\Phi$…
During the past decades, the question of existence and properties of a random attractor of a random dynamical system generated by an S(P)DE has received considerable attention, for example by the work of Gess and R\"ockner. Recently some…
The aim of this paper is to investigate two classical problems related to nilpotent center conditions and bifurcation of limit cycles in switching polynomial systems. Due to the difficulty in calculating the Lyapunov constants of switching…
The aim of this paper is to give a sufficient and necessary condition of the generalized polynomial Li\'enard system with a global center (including linear typer and nilpotent type). Recently, Llibre and Valls [J. Differential Equations,…
Our main result is the complete set of explicit conditions necessary and sufficient for isochronicity of a Hamiltonian system with one degree of freedom. The conditions are presented in terms of Taylor coefficients of the Hamiltonian…
In this paper, we investigate an inverse random source problem concerned with recovering the strength of a random, uncorrelated acoustic source from correlation measurements of emitted time-harmonic acoustic waves. Such problems arise in…
In this paper we prove that any degree $d$ deformation of a generic logarithmic polynomial differential equation with a persistent center must be logarithmic again. This is a generalization of Ilyashenko's result on Hamiltonian differential…
We introduce the relativistic version of the well-known Henon's isochrone spherical models: static spherically symmetrical spacetimes in which all bounded trajectories are isochrone in Henon's sense, i.e., their radial periods do not depend…
We show how the condition of isochronicity can be studied for two dimensional systems in the renormalization group (RG) context. We find a necessary condition for the isochronicity of the Cherkas and another class of cubic systems. Our…
In this article we prove two versions of the Liapunov center theorem for symmetric potentials. We consider a~second order autonomous system $\ddot q(t)=-\nabla U(q(t))$ in the presence of symmetries of a compact Lie group $\Gamma$ acting…
For an attracting periodic orbit (limit cycle) of a deterministic dynamical system, one defines the isochron for each point of the orbit as the cross-section with fixed return time under the flow. Equivalently, isochrons can be…
We present a simpler proof of the existence of an exact number of one or more limit cycles to the Lienard system $\dot{x}=y-F(x) $, $\dot {y}=-g(xt)$, under weaker conditions on the odd functions $F(x) $ and $g(x) $ as compared to those…
We solve the problem of determining under which conditions the monodromy of a vanishing cycle generates the whole homology of a regular fiber for a polynomial $f(x,y)=g(x)+h(y)$ where $h$ and $g$ are polynomials with real coefficients…
The determination of whether a center is isochronous or not is a well-known problem in the qualitative theory of planar systems. In this paper, we explore planar piecewise discontinuous differential systems characterized by a straight…
We prove that the period function of the center at the origin of the $\mathbb{Z}_k$-equivariant differential equation $\dot{z}=iz+a(z\overline{z})^nz^{k+1}, a\ne0,$ is monotonous decreasing for all $n$ and $k$ positive integers, solving a…
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…