Integrability Conditions of a Weak Saddle in a Complex Polynomial Differential System

2020 ◽  
Author(s):  
Jaume Giné ◽  
Claudia Valls
Keyword(s):  
Differential System ◽  
Complex Polynomial ◽  
Polynomial Differential System ◽  
Integrability Conditions ◽  
Weak Saddle

The Liouvillian Integrability of Several Oscillators

2019 ◽  
Vol 29 (05) ◽  
pp. 1950069 ◽  
Author(s):  
Jaume Giné ◽  
Claudia Valls
Keyword(s):  
Differential System ◽  
First Integrals ◽  
Van Der Pol ◽  
Polynomial Differential System ◽  
Liouvillian Integrability ◽  
Cubic Polynomial ◽  
Liouville Integrable

In this paper, we characterize all the Liouvillian first integrals of a cubic polynomial differential system that contains the van der Pol and the Duffing oscillators. It is also shown that the centers correspond to the Liouville integrable cases.

scholarly journals The non-existence, existence and uniqueness of limit cycles for quadratic polynomial differential systems

2017 ◽  
Vol 149 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Jaume Llibre ◽  
Xiang Zhang
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Quadratic Polynomial ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Polynomial Differential System ◽  
Uniqueness Of Limit Cycles

AbstractWe provide sufficient conditions for the non-existence, existence and uniqueness of limit cycles surrounding a focus of a quadratic polynomial differential system in the plane.

scholarly journals On the Generalized Reflective Function of the Three-Degree Polynomial Differential System

2019 ◽  
Vol 2019 ◽  
pp. 1-6
Author(s):  
Jian Zhou ◽  
Shiyin Zhao
Keyword(s):  
Periodic Solutions ◽  
Differential System ◽  
Reflective Function ◽  
Degree Polynomial ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Polynomial Differential System ◽  
Existence Of Periodic Solutions

The structure of the generalized reflective function of three-degree polynomial differential systems is considered in this paper. The generated results are used for discussing the existence of periodic solutions of these systems.

scholarly journals Periodic Orbits Bifurcating from a Nonisolated Zero–Hopf Equilibrium of Three-Dimensional Differential Systems Revisited

2018 ◽  
Vol 28 (05) ◽  
pp. 1850058 ◽  
Author(s):  
Murilo R. Cândido ◽  
Jaume Llibre
Keyword(s):  
Periodic Solutions ◽  
Periodic Orbits ◽  
Differential System ◽  
Three Dimensional ◽  
Averaging Theory ◽  
Chaotic Attractors ◽  
Differential Systems ◽  
Polynomial Differential System

In this paper, we study the periodic solutions bifurcating from a nonisolated zero–Hopf equilibrium in a polynomial differential system of degree two in [Formula: see text]. More specifically, we use recent results of averaging theory to improve the conditions for the existence of one or two periodic solutions bifurcating from such a zero–Hopf equilibrium. This new result is applied for studying the periodic solutions of differential systems in [Formula: see text] having [Formula: see text]-scroll chaotic attractors.

scholarly journals Computing integrability conditions for a cubic differential system

1996 ◽  
Vol 32 (10) ◽  
pp. 99-107 ◽  
Author(s):  
N.G. Lloyd ◽  
J.M. Pearson ◽  
V.A. Romanovsky
Keyword(s):  
Differential System ◽  
Integrability Conditions

scholarly journals Traveling Wave Solutions of a Generalized Camassa-Holm Equation: A Dynamical System Approach

2015 ◽  
Vol 2015 ◽  
pp. 1-19 ◽  
Author(s):  
Lina Zhang ◽  
Tao Song
Keyword(s):  
Differential System ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Single Peak ◽  
Analytical Technique ◽  
Polynomial Differential System ◽  
Holm Equation ◽  
Kink Wave ◽  
Transformation Of Variables ◽  
Planar Polynomial Differential System

We investigate a generalized Camassa-Holm equationC(3,2,2):ut+kux+γ1uxxt+γ2(u3)x+γ3ux(u2)xx+γ3u(u2)xxx=0. We show that theC(3,2,2)equation can be reduced to a planar polynomial differential system by transformation of variables. We treat the planar polynomial differential system by the dynamical systems theory and present a phase space analysis of their singular points. Two singular straight lines are found in the associated topological vector field. Moreover, the peakon, peakon-like, cuspon, smooth soliton solutions of the generalized Camassa-Holm equation under inhomogeneous boundary condition are obtained. The parametric conditions of existence of the single peak soliton solutions are given by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for single peak soliton, kink wave, and kink compacton solutions of theC(3,2,2)equation.

Sign in / Sign up

Export Citation Format

Share Document