On the non-autonomous van der pol equation with large parameter

1972 ◽  
pp. 298-302 ◽  
Author(s):  
N. G. Lloyd
Keyword(s):  
Large Parameter ◽  
Van Der Pol Equation ◽  
Van Der Pol

On a problem of Littlewood concerning Riccati's equation

1969 ◽  
Vol 65 (3) ◽  
pp. 651-662 ◽  
Author(s):  
H. P. F. Swinnerton-Dyer
Keyword(s):  
Riccati Equation ◽  
Initial Condition ◽  
Large Parameter ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Image Position ◽  
Increasing Function ◽  
Parameter Theory

The study of the Riccati equationplays an essential part in the ‘large parameter’ theory of the inhomogeneous van der Pol equation; see for example Littlewood(1), (2). The crucial result is Lemma B of (1), restated and proved as Lemma 5 of (2); for the present paper the relevant parts of it are as follows:Lemma 1. Let z = z(x) be the solution of (1·1) which satisfies the initial condition z = 0 at x = 0, and assume α > 0. Then there is a unique β0 = β0(α) with the property that(i) if β > β0 then z → − ∞ as x → + ∞;(ii) if β < β0 then z → + ∞ at a vertical asymptote x = x0(α,β);(iii) if β = β0 then z ≥ 0 in 0 ≤ x < + ∞ and z = x + β0 + o(1) as x → + ∞.Moreover, β0(α) is a continuous monotone increasing function of α.

On the non-autonomous van der Pol equation with large parameter

1972 ◽  
Vol 72 (2) ◽  
pp. 213-227 ◽  
Author(s):  
N. G. Lloyd
Keyword(s):  
Real Variable ◽  
Large Parameter ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Image Position

We consider the equation.where ø = λt + μ; the dot denotes differentiation with respect to a real variable t (frequently for convenience called ‘time’); b, λ, k, μ are parameters independent of x, ẋ t and b, λ, μ independent of k.

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.

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 Study of fractional order Van der Pol equation

2016 ◽  
Vol 28 (1) ◽  
pp. 55-60 ◽  
Author(s):  
V. Mishra ◽  
S. Das ◽  
H. Jafari ◽  
S.H. Ong
Keyword(s):  
Fractional Order ◽  
Van Der Pol Equation ◽  
Van Der Pol

scholarly journals A generalization of Van der Pol equation

2013 ◽  
Vol 8 ◽  
pp. 1723-1726
Author(s):  
Ana-Magnolia Marin-Ramirez ◽  
Ruben-Dario Ortiz-Ortiz ◽  
Joel-Arturo Rodriguez-Ceballos
Keyword(s):  
Van Der Pol Equation ◽  
Van Der Pol
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