Quadratic Liénard Equations with Quadratic Damping

Dumortier, Freddy; Li, Chengzhi

doi:10.1006/jdeq.1997.3291

Cited by 63 publications

(50 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…As far as we know the Hilbert's numbers H(m, n) are determined only for five cases: H(1, 1) = 0 and H(1, 2) = 1 proved in 1977 by Lins, de Melo and Pugh [15]; H(2, 1) = 1 proved in 1998 by Copell [7]; H(3, 1) = 1 proved in 1990-1996 by Dumortier, Li and Rousseau in [10] and [8]; H(2, 2) = 1 proved in 1997 by Dumortier and Li [9]; and H(1, 3) = 1 proved by Li and Llibre [17] in 2012.…”

Section: Application 1: Generalized Liénard Polynomial Equationmentioning

confidence: 94%

Higher order averaging theory for finding periodic solutions via Brouwer degree

2014

View full text Add to dashboard Cite

Abstract. In this paper we deal with nonlinear differential systems of the formwherefunctions, T -periodic in the first variable, being D an open subset of R n , and ε a small parameter. For such differential systems, which do not need to be of class C 1 , under convenient assumptions we extend the averaging theory for computing their periodic solutions to k-th order in ε. Some applications are also performed.

show abstract

Section: Application 1: Generalized Liénard Polynomial Equationmentioning

confidence: 94%

Higher order averaging theory for finding periodic solutions via Brouwer degree

2014

View full text Add to dashboard Cite

show abstract

“…Let us consider the following example (for more information on those systems we refer to [3]): (m, n)=(2, 2), i.e.…”

Section: Uniform Information By Means Of Family Blow Upmentioning

confidence: 99%

Polynomial Liénard Equations near Infinity

Dumortier¹,

Herssens²

1999

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

We study the behaviour near infinity of the (generalized) Lie nard equationsb k x k providing a complete classification using Poincare compactification as well as Poincare Lyapunov compactification. We show that all the necessary information is contained in a m and b n , except for the so called``center-focus'' problem occuring in case m 2n+1 with m, n odd, and where the behaviour also depends on the value of the other coefficients a i and b j . We also indicate how to take care about obtaining uniform information near infinity. Academic Press

show abstract

“…(v) Dumortier & Rousseau (1990) and Dumortier & Li (1997) proved that H (3, 1) = 1. (vi) In 1997, Dumortier (1997) proved that H (2, 2) = 1. (vii) In 2012, Li & Llibre (2012) proved that H (1, 3) = 1.…”

Section: Introductionmentioning

confidence: 98%

On the number of limit cycles of a class of polynomial differential systems

Llibre

Valls

2012

Proc. R. Soc. A.

View full text Add to dashboard Cite

We study the number of limit cycles of polynomial differential systems of the forṁwhere g 1 , f 1 , g 2 and f 2 are polynomials of a given degree. Note that when g 1 (x) = f 1 (x) = 0, we obtain the generalized polynomial Liénard differential systems. We provide an accurate upper bound of the maximum number of limit cycles that the above system can have bifurcating from the periodic orbits of the linear centreẋ = y,ẏ = −x using the averaging theory of first and second order.

show abstract

Quadratic Liénard Equations with Quadratic Damping

Cited by 63 publications

References 7 publications

Higher order averaging theory for finding periodic solutions via Brouwer degree

Higher order averaging theory for finding periodic solutions via Brouwer degree

Polynomial Liénard Equations near Infinity

On the number of limit cycles of a class of polynomial differential systems

Contact Info

Product

Resources

About