On the critical periods of Liénard systems with cubic restoring forces

Zhengdong Du

International Journal of Mathematics and Mathematical Sciences, 2004, vol. 2004, 1-16

Abstract:

We study local bifurcations of critical periods in the neighborhood of a nondegenerate center of a Liénard system of the form x ˙ = − y + F ( x ) , y ˙ = g ( x ) , where F ( x ) and g ( x ) are polynomials such that deg ( g ( x ) ) ≤ 3 , g ( 0 ) = 0 , and g ′ ( 0 ) = 1 , F ( 0 ) = F ′ ( 0 ) = 0 and the system always has a center at ( 0 , 0 ) . The set of coefficients of F ( x ) and g ( x ) is split into two strata denoted by S I and S I I and ( 0 , 0 ) is called weak center of type I and type II, respectively. By using a similar method implemented in previous works which is based on the analysis of the coefficients of the Taylor series of the period function, we show that for a weak center of type I, at most [ ( 1 / 2 ) deg ( F ( x ) ) ] − 1 local critical periods can bifurcate and the maximum number can be reached. For a weak center of type II, the maximum number of local critical periods that can bifurcate is at least [ ( 1 / 4 ) deg ( F ( x ) ) ] .

Date: 2004
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:949765

DOI: 10.1155/S0161171204402245

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