Results 111 to 120 of about 4,619,884 (143)

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 ...
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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
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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.
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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.
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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
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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
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Bifurcation of Critical Periods from a Quartic Isochronous Center

International Journal of Bifurcation and Chaos, 2014
This paper is focused on the bifurcation of critical periods from a quartic rigidly isochronous center under any small quartic homogeneous perturbations. By studying the number of zeros of the first several terms in the expansion of the period function in ε, it shows that under any small quartic homogeneous perturbations, up to orders 1 and 2 in ε ...
Linping Peng, Zhaosheng Feng
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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
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Generalized isochronous centers for complex systems

Acta Mathematica Sinica, English Series, 2010
The authors consider complex polynomial systems with complex time. Definitions of generalized isochronous centers and period constants are given, and an algorithm is obtained to compute generalized period constants. The method is applied to a class of real cubic Kolmogorov systems.
Wang, Qin Long, Liu, Yi Rong
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Isochronous centers and flat Finsler metrics (I)

Canadian Journal of Mathematics
AbstractThe local structure of rotationally symmetric Finsler surfaces with vanishing flag curvature is completely determined in this paper. A geometric method for constructing such surfaces is introduced. The construction begins with a planar vector field X that depends on two functions of one variable.
Xinhe Mu, Hui Miao, Libing Huang
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