Results 211 to 220 of about 309,932 (262)

Singularities of cubic vector fields

International Journal of Mathematical Education in Science and Technology, 1990
In this paper we prove that the maximum number of elementary singularities with index + 1 for polynomial vector fields of degree 3 on the plane is 6, and we characterize two classes of cubic fields having this maximum number.
A. Urbina, M. Barra, G. Barra
semanticscholar   +2 more sources

The Closed Orbits of a Class of Cubic Vector Fields in ℝ3

Acta Mathematica Sinica, English Series, 2020
In this paper, we investigate the isolated closed orbits of two types of cubic vector fields in ℝ3 by using the idea of central projection transformation, which sets up a bridge connecting the vector field X (x) in ℝ3 with the planar vector fields. We have proved that the cubic vector field in ℝ3 can have two isolated closed orbits or one closed orbit ...
Hongyan Yin, Da Zhou, Xingan Zhang
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Isochronicity into a family of time-reversible cubic vector fields

Applied Mathematics and Computation, 2001
The authors consider planar cubic vector fields with a nondegenerate center at the origin of the \((x,y)\)-plane. Without restriction, the linear part is \((-y,x)\). The aim is to characterize, within a certain subfamily, those vector fields for which the center is isochronous.
J. Chavarriga, I. A. García, J. Giné
semanticscholar   +3 more sources

The integrability conditions for two cubic vector fields

Differential Equations, 2000
The authors give necessary and sufficient conditions for the existence of a first integral of the form \(H(x,y)=xy+\sum_{k\geq 2} F_k(x,y),\) where \(F_k\) are homogeneous polynomials with complex coefficients, for the two complex 6-parameter families of cubic systems given by \[ \dot x=x+axy+bx^2+cy^2+dx^3,\quad \dot y=-y+Axy+By^2+Cx^2+Dy^3, \] where ...
V. Romanovskii, N. L. Shcheglova
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A stratum of cubic vector fields with an integrable saddle and symmetry

Nonlinearity, 1996
In this paper we study a stratum of integrable cubic vector fields with a saddle singularity having symmetry, i.e. symmetric with respect to two axes. Our perspective is from the point of view of invariant algebraic curves of the systems. We study the global geometry of such systems.
Louis-Sebastien Guimond, C. Rousseau
semanticscholar   +3 more sources

Complete study on a bi-center problem for the Z2-equivariant cubic vector fields

Acta Mathematica Sinica, English Series, 2011
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Yi Rong Liu, Ji Li
semanticscholar   +2 more sources

BIFURCATIONS OF CRITICAL PERIODS: CUBIC VECTOR FIELDS IN KAPTEYN'S NORMAL FORM

Quaestiones Mathematicae, 1999
Abstract In this paper, we study the local bifurcations of critical periods in the neighbourhood of a non-degenerate centre of a cubic system in Kapteyn's normal form. We find that this system has no isochronous centres. We show that at most three local critical periods bifurcate from a centre at the origin.
B. Toni
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Bifurcation of Limit Cycles in Cubic Integrable Z2-Equivariant Planar Vector Fields

Qualitative Theory of Dynamical Systems, 2010
Consider the cubic system \[ \begin{aligned} \dot{x}&=-(a+1)\,y+a\,x^{2}y+b\,xy^{2}+c\,y^{3},\\ \dot{y}&=-\tfrac{1}{2}x-d\,y+\tfrac{1}{2}x^{3}+d\,x^{2}y+e\,x y^{2}+f\,y^{3}.\end{aligned} \] The first six focal values at the singular points \((1,0),(-1,0)\) are zero if and only the coefficients satisfy one of 11 sets of conditions. The first set implies
P. Yu, Man-Un Han, Jibin Li
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Limit cycles of cubic polynomial vector fields via the averaging theory

Nonlinear Analysis: Theory, Methods & Applications, 2007
This paper deals with the number of limit cycles that can bifurcate from the period annulus surrounding the origin of real, planar, cubic differential systems. The method used to do this study is the averaging method at first order in the perturbation parameter \(\varepsilon\). In particular the considered systems are of the form: \[ \dot{x} = -y f(x,y)
J. Giné, J. Llibre
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Darboux transformation, generalized Darboux transformation and vector breather solutions for a coupled variable-coefficient cubic-quintic nonlinear Schrödinger system in a non-Kerr medium, twin-core nonlinear optical fiber or waveguide

Waves in Random and Complex Media, 2021
Non-Kerr media, twin-core nonlinear optical fibers and waveguides are widely applied in optical fiber communications. In order to model the effects of quintic nonlinearity for the ultrashort optical pulse propagation in a non-Kerr medium, twin-core ...
Meng Wang, Bo Tian
semanticscholar   +1 more source