On the Maximum Number of Limit Cycles of a Class of Generalized Liénard Differential Systems

Int. J. Bifurcation Chaos

Valls

2013

Self Cite

We study the number of limit cycles of the polynomial differential systems of the form [Formula: see text] where g1(x) = εg11(x) + ε2g12(x) + ε3g13(x), g2(x) = εg21(x) + ε2g22(x) + ε3g23(x) and f(x) = εf1(x) + ε2 f2(x) + ε3 f3(x) where g1i, g2i, f2i have degree k, m and n respectively for each i = 1, 2, 3, and ε is a small parameter. Note that when g1(x) = 0 we obtain the generalized Liénard polynomial differential systems. We provide an upper bound of the maximum number of limit cycles that the previous differential system can have bifurcating from the periodic orbits of the linear center ẋ = y, ẏ = -x using the averaging theory of third order.

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 97%

“…In [Alavez-Ramirez et al, 2012] the authors took the sufficient conditions b 2i+1,1 = 0 and a 2j,2 = 0,…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

“…, m, thus obtaining an upper bound for the number of limit cycles. However a close look to F 20 (r) obtained in [Alavez-Ramirez et al, 2012] imply that it is only necessary to take…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

Section: Proof Of Theorem 13mentioning

confidence: 99%

“…It was proved in [Alavez-Ramirez et al, 2012] that using the integrals of the Appendix, or in [Llibre & Valls, 2011] that y 1 = y 1 (θ, r) is equal to…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

See 3 more Smart Citations

On the Number of Limit Cycles for a Generalization of Liénard Polynomial Differential Systems

Int. J. Bifurcation Chaos

Valls

2013

Self Cite

Limit cycles for a generalization of polynomial Liénard differential systems

Chaos, Solitons & Fractals

Valls

2013

Self Cite

We study the number of limit cycles of the polynomial differential systems of the forṁ x = y − f 1 (x)y,ẏ = −x − g 2 (x) − f 2 (x)y, where f 1 (x) = εf 11 (x) + ε 2 f 12 (x) + ε 3 f 13 (x), g 2 (x) = εg 21 (x) + ε 2 g 22 (x) + ε 3 g 23 (x) and f 2 (x) = εf 21 (x)+ε 2 f 22 (x)+ε 3 f 23 (x) where f 1i , f 2i and g 2i have degree l, n and m respectively for each i = 1, 2, 3, and ε is a small parameter. Note that when 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 this differential system can have bifurcating from the periodic orbits of the linear centerẋ = y,ẏ = −x using the averaging theory of third order.

Limit cycles of generalized Liénard polynomial differential systems via averaging theory

García

Chaos, Solitons & Fractals

Río

2014

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Using the averaging theory of first and second order we study the maximum number of limit cycles of the polynomial differential systemṡwhich bifurcate from the periodic orbits of the linear centerẋ = y,ẏ = −x.Here ε is a small parameter. If the degrees of the polynomials p 1 , p 2 , q 1 and q 2 is n, then we prove that this maximum number is [n/2] using the averaging theory of first order, where [•] denotes the integer part function; and this maximum number is at most n using the averaging theory of second order.