Perturbation of a period annulus bounded by a heteroclinic loop connecting two hyperbolic saddles
The author considers the perturbed Liénard system \[ \dot{x} = y\;,\;\dot{y}= - {{1}\over{4}}x(x^2-1)(x^2+2)^2 + \varepsilon (\alpha_0+\alpha_1x^2+\alpha_2x^4+\alpha_3x^6)y, \] with \(0<|\varepsilon|\ll 1\), \(\alpha_i \in \mathbb{R}\), \(i=0,1,2,3\). He considers the weak 16th Hilbert problem, thus analyzing the maximal number of zeros of the associated abelian integral. The main result reads as follows: for all \((\alpha_0,\alpha_1,\alpha_2,\alpha_3)\in \mathbb{R}^4\), the associated abelian integral has at most four zeros in the interval \((0,11/32)\).
scientific article; zbMATH DE number 6855272
scientific article