Limit cycles near an eye-figure loop in some polynomial Liénard systems
The Hamiltonian system \[ \dot{x} = y, \quad \dot{y} = (x-a)(x-b)^2(x-c)(x-d)^2(x-e) \] with the Hamiltonian function \[ H(x,y) = \frac{1}{2}y^2+G(x),\;\;\;G(x) = -\int_0^{x}(t-a)(t-b)^2(t-c)(t-d)^2(t-e)dt \] for \( a < b < c < d < e \) can be written as the system \[ \dot{x} =y, \;\;\;\dot{y} =x (x^2-1)(x^2-\bar{d}^2)^2 \] where \(0 < \bar{d} < 1\) assuming \(G(a) =G(e)\) and \(G(b) =G(d)\). The authors prove a lower bound on the number of limit cycles of the near-Hamiltonian Liénard system \[ \dot{x} =y, \;\;\;\dot{y}=x (x^2-1)(x^2-\frac{1}{4})^2 + \epsilon f(x,\delta)y \] where \[ f(x,\delta) = \sum_{i=0}^ka_ix^{2i}, \;\;a_i\in \mathbb{R} \] for \(k =2,3,4, \text{ and } 5\), assuming \(0 < \epsilon \ll 1\). The proof is based on the results developed by \textit{M. Han} and collaborators in the series of papers [Int. J. Bifurcation Chaos Appl. Sci. Eng. 19, No. 12, 4117--4130 (2009; Zbl 1183.34054); Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 9, 2948--2965 (2011; Zbl 1218.34045); J. Differ. Equations 246, No. 1, 129--163 (2009; Zbl 1168.34029); J. Dyn. Differ. Equations 20, No. 4, 923--944 (2008; Zbl 1165.34016)].
scientific article; zbMATH DE number 6756248
scientific article