Results 111 to 120 of about 1,422 (148)

Centers and Isochronous Centers of Liénard Systems

Differential Equations, 2019
Holomorphic Liénard systems are studied. The authors give necessary and sufficient conditions for the existence of a center and an isochronous center which are obtained without calculating the focus quantities and the isochronicity constants.
Amel'kin, V. V., Rudenok, A. E.
openaire   +1 more source

Isochronous Centers and Isochronous Functions

Acta Mathematicae Applicatae Sinica, 2002
The author investigates the isochronous centers of two classes of planar systems of ordinary differential equations: 1) Liénard systems of the form \((\dot x)=y-F(x),(\dot y)=-g(x)\), 2) Hamiltonian systems of the form \((\dot x)=-g(y)\), \((\dot y)=f(x)\), with emphasis on the case when the functions \(g\) or \(f\) are isochronous. For the first class
exaly   +3 more sources

Isochronicity of centers at a center manifold

AIP Conference Proceedings, 2012
For a three dimensional system with a center manifold filled with closed trajectories (corresponding to periodic solutions of the system) we give criteria on the coefficients of the system to distinguish between the cases of isochronous and non-isochronous oscillations. Bifurcations of critical periods of the system are studied as well.
Brigita Ferčec, Matej Mencinger
openaire   +1 more source

Rational Liénard Systems with a Center and an Isochronous Center

Differential Equations, 2020
The following Liénard system \[ \Dot{x}=-y,\quad\Dot{y}=f(x)+yg(x)\tag{1} \] is considered with rational functions \(f\) and \(g\), where the functions \(f\) and \(g\) are linearly independent and holomorphic, and \(f(0)=g(0)=0\) and \(f'(0)=1\). Firstly, the definition of degree of an element \(g(x)/h(x)\) of the field \(k(x)\) and the definition of ...
openaire   +2 more sources

Center and isochronous center problems for quasi analytic systems

Acta Mathematica Sinica, English Series, 2008
Consider the planar quasi-analytic systems \[ \begin{aligned} {dx\over dt} &=\delta x- y+ \sum^\infty_{k=2} (x^2+ y^2)^{(k-1)(\lambda- 1)/2}X_k(x,y),\\ {dy\over dt} &= x+\delta y+ \sum^\infty_{k=2} (x^2+ y^2)^{(k-1)(\lambda- 1)/2} Y_k(x, y),\end{aligned} \] where \[ \begin{aligned} X_k(x, y) &= \sum_{\alpha+\beta= k} A_{\alpha\beta} x^\alpha y^\beta,\\
Liu, Yi Rong, Li, Ji Bin
exaly   +2 more sources

Periodic perturbations of an isochronous center

Qualitative Theory of Dynamical Systems, 2002
The author discusses the possibility of producing resonance in a nonlinear isochronous center. In some cases it is shown than one can find periodic forcings (with the same period of the center) such that the solutions of the perturbed equation are unbounded.
openaire   +2 more sources

Isochronous Centers in Planar Polynomial Systems

SIAM Journal on Mathematical Analysis, 1997
The authors consider the following planar system \[ \dot x = P(x,y),\quad \dot y = Q(x,y),\qquad (x,y)\in \mathbb{R}^2 \tag{1} \] where \(P\) and \(Q\) are polynomials in \(x\) and \(y\). Let the system (1) have a center, and let \(T\) be a period-function, i.e.
Christopher, C. J., Devlin, J.
exaly   +2 more sources

Number of Critical Periods for Perturbed Rigidly Isochronous Centers

International Journal of Bifurcation and Chaos, 2016
This paper deals with the bifurcation of critical periods from a rigidly quartic isochronous center. It shows that under any small homogeneous perturbation of degree four, up to any order in [Formula: see text], there are at most two critical periods bifurcating from the periodic orbits of the unperturbed system, and the upper bound is sharp.
Lianghaolong Lu, Linping Peng, Zhaosheng Feng   +2 more
openaire   +2 more sources

The isochronous centers for Kukles homogeneous system of degree nine

Applied Mathematics Letters, 2021
The authors consider Kukles homogeneous systems \begin{align*} & \dot{x}=-y, \\ & \dot{y}=x+{{Q}_{n}}(x,y), \\ \end{align*} where \({{Q}_{n}}(x,y)\) is a homogeneous polynomial of degree \(n\). The authors point out gaps in the proof of Conjecture 1 and 2 for considered system in papers [\textit{J. Giné} et al., Bull. Lond. Math. Soc. 47, No.
Lina Guo, Changjian Liu
openaire   +2 more sources

ON THE NUMBER OF ZEROS OF ABELIAN INTEGRAL FOR A CUBIC ISOCHRONOUS CENTER

International Journal of Bifurcation and Chaos, 2012
In this paper, we study the number of limit cycles that bifurcate from the periodic orbits of a cubic reversible isochronous center under cubic perturbations. It is proved that in this situation the least upper bound for the number of zeros (taking into account the multiplicity) of the Abelian integral associated with the system is equal to four ...
Kuilin Wu, Yunlin Zhao
openaire   +1 more source