Small-amplitude limit cycles in polynomial Liénard systems (Q1914378)

For the Liénard equation \(d^2x/dt^2 + f(x) dx/dt + g(x) = 0\) or its equivalent two dimensional system (1) \(dx/dt = y\), \(dy/dt = - g(x) - yf(x)\), assume that \(f\) and \(g\) are polynomials with \(f(0) = g(0) = 0\), \(g'(0) > 0\). The origin is then either a fine focus or a centre. The paper is concerned with the order of such a fine focus. Let \({\mathcal L} (n,m)\) denote the class of Liénard systems in which \(g'(0) > 0\) and degrees of \(f\) and \(g\) are \(n\) and \(m\), respectively, and let \(H(n,m)\) be the maximum order of a fine focus for systems in \({\mathcal L} (n,m)\). Each system (1) in \({\mathcal L} (n,m)\) can be identified by the pair \((f,g)\). The following results are obtained by the authors. Theorem. Let \(\varphi (f,g)\) be the order of the fine focus at the origin for (1), and \({\mathcal L}_0 (n,m) = \{(f,g) \mid (f,g) \in {\mathcal L} (n,m)\) and \(f'(0) > 0\}\), \(H_0 (n,m) = \max \{\varphi (f,g) \mid (f,g) \in {\mathcal L}_0 (n,m)\}\). (i) If \((f,g) \in {\mathcal L}_0 (n,m)\), then \(\varphi (f,g) = \varphi (g,f)\). (ii) \([{1 \over 2} (n + m - 1)] \leq H_0 (n,m) \leq [{1 \over 2} nm]\). (iii) For \(n,m > 1\), \(H_0 (n,m) = H_0 (m,n)\). (iv) For \(n \geq m\), we have \(H(n,m) =H_0 (n,m) \leq H(m,n)\).