An upper bound for the number of small-amplitude limit cycles in non-smooth Liénard system (Q6577351)

The Liénard equation is defined by the second-order nonlinear differential equation\N\[\N\tag{1} \ddot{x}+f(x)\dot{x}+g(x)=0,\N\]\Nwhere \(f\) and \(g\) are continuous, \(xg(x)>0\) when \(x\neq 0\), \(f(0)<0\). In order to study the equation \((1)\) one can consider \(G(x)=\int_{0}^{x}g(\xi)d\xi\), \(F(x)=\int_{0}^{x}f(\xi)d\xi\) and we rewrite the Liénard equation as\N\[\N(\dot{x}, \dot{y})=(y-F(x),-g(x)),\N\]\Nor, more general, as\N\[\N\tag{2} (\dot{x}, \dot{y})=(p(y)-F(x,a),-g(x)),\N\]\Ncalled Liénard system, where \(F\), \(g\) and \(p\) are \(C^{\infty}\) functions near the origin with \(p(0)=0\), \(g(0)=0\), \(g'(0)>0\), \(p'(0)>0\) and \(F(0,a)=0\), \(a\in \mathbb R^s.\) For more details see [\textit{Y. Tian} and \textit{M. Han}, J. Differ. Equations 251, No. 4--5, 834--859 (2011; Zbl 1228.34059)].\N\NIn this paper, the author considers a piecewise Liénard system defined by\N\[\N(\dot{x},\dot{y})=\left\lbrace \begin{array}{rcl} X^+ \text{ for } &x>0,\\\NX^-\text{ for }& x\leq 0, \end{array} \right.\N\]\Nwhere \(X^+=(p(y)-F^+(x,a),-g^+(x))\) and \(X^-=(p(y)-F^-(x,a),-g^-(x))\), satisfying the additional condition \((F^{\pm}(0,a_0))^2-4p(0)(g^{\pm})'(0)<0\), and investigates the small-amplitude limit cycles generated by Hopf bifurcation. In order to determine the upper bound of the number of small amplitude limit cycles for a generic non-smooth Liénard system, the author uses the Picard-Fuchs equation. The Picard-Fuchs equations method can simplify the investigation of limit cycles of discontinuous differential systems in general, this method was used in [\textit{J. Yang} and \textit{L. Zhao}, J. Differ. Equations 264, No. 9, 5734--5757 (2018; Zbl 1390.34082)]. Furthermore, an example is presented to illustrate the results.