scholarly journals Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields

2018 ◽  
Vol 16 (1) ◽  
pp. 1204-1217
Author(s):  
Primitivo B. Acosta-Humánez ◽  
Alberto Reyes-Linero ◽  
Jorge Rodriguez-Contreras
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Vector Fields ◽  
Qualitative Theory ◽  
Parametric Family ◽  
Liénard Equation ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Qualitative Approaches ◽  
Lienard Equation

AbstractIn this paper we study a particular parametric family of differential equations, the so-called Linear Polyanin-Zaitsev Vector Field, which has been introduced in a general case in [1] as a correction of a family presented in [2]. Linear Polyanin-Zaitsev Vector Field is transformed into a Liénard equation and, in particular, we obtain the Van Der Pol equation. We present some algebraic and qualitative results to illustrate some interactions between algebra and the qualitative theory of differential equations in this parametric family.

scholarly journals Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1

2019 ◽  
Vol 17 (1) ◽  
pp. 1220-1238
Author(s):  
Jorge Rodríguez-Contreras ◽  
Primitivo B. Acosta-Humánez ◽  
Alberto Reyes-Linero
Keyword(s):  
Differential Equations ◽  
Point Of View ◽  
Parametric Family ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Algebraic Point ◽  
The Family

Abstract The aim of this paper is the analysis, from algebraic point of view and singularities studies, of the 5-parametric family of differential equations $$\begin{array}{} \displaystyle yy'=(\alpha x^{m+k-1}+\beta x^{m-k-1})y+\gamma x^{2m-2k-1}, \quad y'=\frac{dy}{dx} \end{array}$$ where a, b, c ∈ ℂ, m, k ∈ ℤ and $$\begin{array}{} \displaystyle \alpha=a(2m+k) \quad \beta=b(2m-k), \quad \gamma=-(a^2mx^{4k}+cx^{2k}+b^2m). \end{array}$$ This family is very important because include Van Der Pol equation. Moreover, this family seems to appear as exercise in the celebrated book of Polyanin and Zaitsev. Unfortunately, the exercise presented a typo which does not allow to solve correctly it. We present the corrected exercise, which corresponds to the title of this paper. We solve the exercise and afterwards we make algebraic and of singularities studies to this family of differential equations.

Systems of differential equations that are competitive or cooperative. VI: A localCrClosing Lemma for 3-dimensional systems

1991 ◽  
Vol 11 (3) ◽  
pp. 443-454 ◽  
Author(s):  
Morris W. Hirsch
Keyword(s):  
Vector Field ◽  
Positive Integer ◽  
Differential Equations ◽  
Vector Fields ◽  
3 Dimensional ◽  
Planar Surfaces ◽  
Systems Of Differential Equations ◽  
Nonwandering Point

AbstractFor certainCr3-dimensional cooperative or competitive vector fieldsF, whereris any positive integer, it is shown that for any nonwandering pointp, every neighborhood ofFin theCrtopology contains a vector field for whichpis periodic, and which agrees withFoutside a given neighborhood ofp. The proof is based on the existence of invariant planar surfaces throughp.

On Integrating Factors and a Perturbation Method

1995 ◽  
Author(s):  
W. T. van Horssen
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Ordinary Differential Equations ◽  
First Integrals ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Order Ordinary Differential Equation ◽  
First Order ◽  
Integrating Factors ◽  
Complete Solutions

Abstract In this paper the fundamental concept (due to Euler, 1734) of how to make a first order ordinary differential equation exact by means of integrating factors, is extended to n-th order (n ≥ 2) ordinary differential equations and to systems of first order ordinary differential equations. For new classes of differential equations first integrals or complete solutions can be constructed. Also a perturbation method based on integrating factors can be developed. To show how this perturbation method works the method is applied to the well-known Van der Pol equation.

A weight-homogenous condition to the real Jacobian conjecture in

2021 ◽  
pp. 1-9
Author(s):  
Francisco Braun ◽  
Claudia Valls
Keyword(s):  
Differential Equations ◽  
Vector Fields ◽  
Qualitative Theory ◽  
Hamiltonian Vector ◽  
Jacobian Conjecture ◽  
Local Diffeomorphism ◽  
The Real ◽  
Hamiltonian Vector Fields ◽  
Real Linear

Abstract It is known that a polynomial local diffeomorphism $(f,\, g): {\mathbb {R}}^{2} \to {\mathbb {R}}^{2}$ is a global diffeomorphism provided the higher homogeneous terms of $f f_x+g g_x$ and $f f_y+g g_y$ do not have real linear factors in common. Here, we give a weight-homogeneous framework of this result. Our approach uses qualitative theory of differential equations. In our reasoning, we obtain a result on polynomial Hamiltonian vector fields in the plane, generalization of a known fact.

scholarly journals Application of the Hori Method in the Theory of Nonlinear Oscillations

2012 ◽  
Vol 2012 ◽  
pp. 1-32
Author(s):  
Sandro da Silva Fernandes
Keyword(s):  
Differential Equations ◽  
Generating Function ◽  
Nonlinear Oscillations ◽  
System Of Differential Equations ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Identity Transformations ◽  
Hori Method ◽  
New System ◽  
Made In

Some remarks on the application of the Hori method in the theory of nonlinear oscillations are presented. Two simplified algorithms for determining the generating function and the new system of differential equations are derived from a general algorithm proposed by Sessin. The vector functions which define the generating function and the new system of differential equations are not uniquely determined, since the algorithms involve arbitrary functions of the constants of integration of the general solution of the new undisturbed system. Different choices of these arbitrary functions can be made in order to simplify the new system of differential equations and define appropriate near-identity transformations. These simplified algorithms are applied in determining second-order asymptotic solutions of two well-known equations in the theory of nonlinear oscillations: van der Pol equation and Duffing equation.

scholarly journals Complete lift of vector fields and sprays to T∞M

2015 ◽  
Vol 12 (10) ◽  
pp. 1550113 ◽  
Author(s):  
Ali Suri ◽  
Somaye Rastegarzadeh
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Dimensional Manifold ◽  
Closed Curves ◽  
Infinite Dimensional ◽  
Integral Curves ◽  
Complete Lift ◽  
Infinite Dimensional Manifold

In this paper for a given Banach, possibly infinite dimensional, manifold M we focus on the geometry of its iterated tangent bundle TrM, r ∈ ℕ ∪ {∞}. First we endow TrM with a canonical atlas using that of M. Then the concepts of vertical and complete lifts for functions and vector fields on TrM are defined which they will play a pivotal role in our next studies i.e. complete lift of (semi)sprays. Afterward we supply T∞M with a generalized Fréchet manifold structure and we will show that any vector field or (semi)spray on M, can be lifted to a vector field or (semi)spray on T∞M. Then, despite of the natural difficulties with non-Banach modeled manifolds, we will discuss about the ordinary differential equations on T∞M including integral curves, flows and geodesics. Finally, as an example, we apply our results to the infinite-dimensional case of manifold of closed curves.

scholarly journals On constructing single-input non-autonomous systems of full rank

2018 ◽  
Keyword(s):  
Vector Field ◽  
Nonlinear System ◽  
Differential Equations ◽  
Autonomous System ◽  
Vector Fields ◽  
Autonomous Systems ◽  
Full Rank ◽  
System Of Differential Equations ◽  
Single Input ◽  
Class C

For a nonlinear system of differential equations $\dot x=f(x)$, a method of constructing a system of full rank $\dot x=f(x)+g(x)u$ is studied for vector fields of the class $C^k$, $1\le k<\infty$, in the case when $f(x)\not=0$. A method for constructing a non-autonomous system of full rank is proposed in the case when the vector field $f(x)$ can vanish.

scholarly journals ON THE PERIODIC SOLUTIONS OF THE LIENARD EQUATION

2020 ◽  
pp. 7-19
Author(s):  
Abdula Abdullaev ◽  
◽  
Anna Savochkina ◽  
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Operator Equation ◽  
Boundary Value ◽  
Linear Part ◽  
Functional Differential Equations ◽  
Liénard Equation ◽  
Functional Differential ◽  
Lienard Equation

Mathematical modeling of many problems of natural science leads to the need to study quasi-linear boundary value problems for functional differential equations with a linear part that is not uniquely solvable for all right-hand parts. The specificity of such problems is that the corresponding linear operator is not reversible. In the literature, such boundary value problems are usually called resonant. Since the 70s of the last century, the development of methods for studying resonant boundary value problems considered as a single operator equation has begun. A very important area of research from the point of view of applications is the application of General statements to the study of periodic boundary value problems for functional differential equations. The existence problem is considered ω - a periodic solution of the Lienard equation with a deviating argument of the form It is assumed that the function p ( t ) is measurable and Using an approach based on the application of theoretical existence for a quasilinear operator equation, sufficient conditions can be obtained in the work, at least one ω - a periodic solution must correspond to the equations. The obtained result refines some well-known results for the Lienard equations. Execution conditions decisions do not affect the existence of solutions.

scholarly journals A High-Order Melnikov Method for Heteroclinic Orbits in Planar Vector Fields and Heteroclinic Persisting Perturbations

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Yi Zhong
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Heteroclinic Orbit ◽  
Melnikov Method ◽  
High Order ◽  
Heteroclinic Orbits ◽  
Van Der Pol ◽  
Planar Vector Fields ◽  
Planar Vector

This work extends the high-order Melnikov method established by FJ Chen and QD Wang to heteroclinic orbits, and it is used to prove, under a certain class of perturbations, the heteroclinic orbit in a planar vector field that remains unbroken. Perturbations which have this property together form the heteroclinic persisting space. The Van der Pol system is analysed as an application.

scholarly journals LAGRANGIAN FLOWS FOR VECTOR FIELDS WITH GRADIENT GIVEN BY A SINGULAR INTEGRAL

2013 ◽  
Vol 10 (02) ◽  
pp. 235-282 ◽  
Author(s):  
FRANÇOIS BOUCHUT ◽  
GIANLUCA CRIPPA
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Singular Integral ◽  
Vector Fields ◽  
Transport Equations ◽  
Well Posedness ◽  
Quantitative Stability ◽  
Quantitative Estimates

We prove quantitative estimates on flows of ordinary differential equations with vector field with gradient given by a singular integral of an L1 function. Such estimates allow to prove existence, uniqueness, quantitative stability and compactness for the flow, going beyond the BV theory. We illustrate the related well-posedness theory of Lagrangian solutions to the continuity and transport equations.

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