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.

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.

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.

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 Asymptotic Differential Algebra and Model Theory of Transseries

2017 ◽  
Author(s):  
Matthias Aschenbrenner ◽  
Lou van den Dries ◽  
Joris van der Hoeven
Keyword(s):  
Differential Equations ◽  
Vector Fields ◽  
Algebraic Closure ◽  
Differential Algebra ◽  
Quantifier Elimination ◽  
Point Of View ◽  
Algebraic Point ◽  
Differential Field ◽  
Algebraic Setting ◽  
Differential Fields

Asymptotic differential algebra seeks to understand the solutions of differential equations and their asymptotics from an algebraic point of view. The differential field of transseries plays a central role in the subject. Besides powers of the variable, these series may contain exponential and logarithmic terms. Over the last thirty years, transseries emerged variously as super-exact asymptotic expansions of return maps of analytic vector fields, in connection with Tarski's problem on the field of reals with exponentiation, and in mathematical physics. Their formal nature also makes them suitable for machine computations in computer algebra systems. This book validates the intuition that the differential field of transseries is a universal domain for asymptotic differential algebra. It does so by establishing in the realm of transseries a complete elimination theory for systems of algebraic differential equations with asymptotic side conditions. Beginning with background chapters on valuations and differential algebra, the book goes on to develop the basic theory of valued differential fields, including a notion of differential-henselianity. Next, H-fields are singled out among ordered valued differential fields to provide an algebraic setting for the common properties of Hardy fields and the differential field of transseries. The study of their extensions culminates in an analogue of the algebraic closure of a field: the Newton–Liouville closure of an H-field. This paves the way to a quantifier elimination with interesting consequences.

Periodic Motion of the van der Pol Equation with Impulsive Effect

2015 ◽  
Vol 25 (09) ◽  
pp. 1550119 ◽  
Author(s):  
Shasha Zheng ◽  
Xilin Fu
Keyword(s):  
Fixed Points ◽  
Periodic Motion ◽  
Point Of View ◽  
Impulsive Effect ◽  
Switching Systems ◽  
Periodic Motions ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Mapping Structures ◽  
Impulsive Model

In this paper, we investigate the dynamics of the periodic motion for switched van der Pol equation with impulsive effect, utilizing the theory of mapping dynamics in switching systems. For the optimized problem, we consider such impulsive dynamical model as switched system and analyze its features from a discontinuous point of view. Then, conceptions of switching sets as well as discrete mappings are briefly reviewed. By constructing generic mappings, we analyze the flow's periodic behaviors from the perspective of mapping structures. Finally, we apply our analysis and criterion to a specific impulsive model at fixed points and the periodic motions with impulse to the boundary are illustrated.

scholarly journals Hyperbolastic Models from a Stochastic Differential Equation Point of View

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1835
Author(s):  
Antonio Barrera ◽  
Patricia Román-Román ◽  
Francisco Torres-Ruiz
Keyword(s):  
Differential Equation ◽  
Simulated Data ◽  
Real Data ◽  
Point Of View ◽  
Likelihood Method ◽  
Numerical Resolution ◽  
The Family ◽  
The Common ◽  
Linear Differential ◽  
Mean Function

A joint and unified vision of stochastic diffusion models associated with the family of hyperbolastic curves is presented. The motivation behind this approach stems from the fact that all hyperbolastic curves verify a linear differential equation of the Malthusian type. By virtue of this, and by adding a multiplicative noise to said ordinary differential equation, a diffusion process may be associated with each curve whose mean function is said curve. The inference in the resulting processes is presented jointly, as well as the strategies developed to obtain the initial solutions necessary for the numerical resolution of the system of equations resulting from the application of the maximum likelihood method. The common perspective presented is especially useful for the implementation of the necessary procedures for fitting the models to real data. Some examples based on simulated data support the suitability of the development described in the present paper.

scholarly journals On Periodic Perturbations of Asymmetric Duffing–Van-der-Pol Equation

2014 ◽  
Vol 24 (05) ◽  
pp. 1450061 ◽  
Author(s):  
Albert D. Morozov ◽  
Olga S. Kostromina
Keyword(s):  
Saddle Point ◽  
Limit Cycles ◽  
Small Neighborhood ◽  
Integrable Equation ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Periodic Perturbations ◽  
Autonomous Equation ◽  
Nonautonomous Equation ◽  
Time Periodic

Time-periodic perturbations of an asymmetric Duffing–Van-der-Pol equation close to an integrable equation with a homoclinic "figure-eight" of a saddle are considered. The behavior of solutions outside the neighborhood of "figure-eight" is studied analytically. The problem of limit cycles for an autonomous equation is solved and resonance zones for a nonautonomous equation are analyzed. The behavior of the separatrices of a fixed saddle point of the Poincaré map in the small neighborhood of the unperturbed "figure-eight" is ascertained. The results obtained are illustrated by numerical computations.

A Novel General and Robust Method Based on NAOP for Solving Nonlinear Ordinary Differential Equations and Partial Differential Equations by Cellular Neural Networks

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
Jean Chamberlain Chedjou ◽  
Kyandoghere Kyamakya
Keyword(s):  
Neural Networks ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Cellular Neural Networks ◽  
Nonlinear Ordinary Differential Equations ◽  
Van Der Pol ◽  
Partial Differential ◽  
Nonlinear Odes ◽  
First Order

This paper develops and validates through a series of presentable examples, a comprehensive high-precision, and ultrafast computing concept for solving nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN). The core of this concept is a straightforward scheme that we call "nonlinear adaptive optimization (NAOP),” which is used for a precise template calculation for solving nonlinear ODEs and PDEs through CNN processors. One of the key contributions of this work is to demonstrate the possibility of transforming different types of nonlinearities displayed by various classical and well-known nonlinear equations (e.g., van der Pol-, Rayleigh-, Duffing-, Rössler-, Lorenz-, and Jerk-equations, just to name a few) unto first-order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN templates. Furthermore, in the case of PDE solving, the same concept also allows a mapping unto first-order CNN cells while considering one or even more nonlinear terms of the Taylor's series expansion generally used in the transformation of a PDE in a set of coupled nonlinear ODEs. Therefore, the concept of this paper does significantly contribute to the consolidation of CNN as a universal and ultrafast solver of nonlinear ODEs and/or PDEs. This clearly enables a CNN-based, real-time, ultraprecise, and low-cost computational engineering. As proof of concept, two examples of well-known ODEs are considered namely a second-order linear ODE and a second order nonlinear ODE of the van der Pol type. For each of these ODEs, the corresponding precise CNN templates are derived and are used to deduce the expected solutions. An implementation of the concept developed is possible even on embedded digital platforms (e.g., field programmable gate array (FPGA), digital signal processor (DSP), graphics processing unit (GPU), etc.). This opens a broad range of applications. Ongoing works (as outlook) are using NAOP for deriving precise templates for a selected set of practically interesting ODEs and PDEs equation models such as Lorenz-, Rössler-, Navier Stokes-, Schrödinger-, Maxwell-, etc.

scholarly journals CONFLICTS AND FAIR TESTING

2006 ◽  
Vol 17 (04) ◽  
pp. 797-813 ◽  
Author(s):  
ROBI MALIK ◽  
DAVID STREADER ◽  
STEVE REEVES
Keyword(s):  
Discrete Event Systems ◽  
Process Algebra ◽  
Discrete Event ◽  
Denotational Semantics ◽  
Point Of View ◽  
Algebraic Point ◽  
Event Systems ◽  
Common Task ◽  
Testing Theory ◽  
Process Algebraic

This paper studies conflicts from a process-algebraic point of view and shows how they are related to the testing theory of fair testing. Conflicts have been introduced in the context of discrete event systems, where two concurrent systems are said to be in conflict if they can get trapped in a situation where they are waiting or running endlessly, forever unable to complete their common task. In order to analyse complex discrete event systems, conflict-preserving notions of refinement and equivalence are needed. This paper characterises an appropriate refinement, called the conflict preorder, and provides a denotational semantics for it. Its relationship to other known process preorders is explored, and it is shown to generalise the fair testing preorder in process-algebra for reasoning about conflicts in discrete event systems.

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