Small amplitude limit cycles for the polynomial Liénard system
This paper gives an estimation for the maximal number of small amplitude limit cycles bifurcating from the singular point \(x=0\) of focus type for the Liénard equation \[ \ddot x + f(x) \dot x + g(x) = 0, \quad g(0)=0, \;\, g'(0) > 0, \tag{1} \] where \(f\) and \(g\) are real polynomials of degree \(m\) and \(n\), respectively. The authors obtain estimates quadratic in \(m, n\) improving the previous bounds. The Liénard equation (1) is naturally related to the vector field \[ \dot x = y - F(x), \quad \dot y = -G'(x), \quad x, y \in {\mathbb R}^1, \tag{2} \] where \(F\) and \(G\) are real polynomials of degree \(m+1\) and \(n+1\) respectively connected with \(f\) and \(g\) via the relations \(F'(x)=f(x)\), \(F(0)=0\) and \(G'(x)=g(x)\). The vector field (2) can be analytically transformed into the orbitally equivalent one: \[ \dot u = y - \Phi(u), \quad \dot y = -2u, \quad \Phi(u) = c_1u + c_2u^2 + c_3u^3 + \cdots. \] Here, the series \(X = \Phi(Y^{\frac{1}{2}}) = c_1Y^{\frac{1}{2}} + c_2 Y + c_3 Y^{\frac{3}{2}} +\cdots\) is the Puiseux expansion at the origin of the algebraic curve \(C: \;X=F(x), \;Y=G(x)\) on the \((X,Y)\)-plane. Using methods from complex algebraic geometry, the authors estimate the number of double points of the curve \(C\) and prove that the maximal number of double points is \(\delta_{\max} = \frac{1}{2} (mn - \text{ gcd} (m+1,n+1) + 1)\). Let \(\widehat H\) be the maximal number of limit cycles of the vector field (2) bifurcating from the origin, and \(H^*\) be the maximal cyclicity of the origin. Let \(\widehat H_{\mathbb C}\) and \(H^*_{\mathbb C}\) be the similar notions in the complex sense. There are simple relations: \(\widehat H \leq H^* \leq H^*_{\mathbb C} = \widehat H_{\mathbb C}\), and the general estimation \(H^*_{\mathbb C} \leq \frac{1}{2} mn\). In the present paper the authors obtain the following general estimates: \[ H^*_{\mathbb C} \leq \delta_{\max}-1, \quad H^*_{\mathbb C} \leq \tfrac{1}{4} (mn + 3m+3n+1) \tag{3} \] for all \(m,n \geq 2\) .
scientific article; zbMATH DE number 5362102
scientific article