Invariant algebraic curves and hyperelliptic limit cycles of Liénard systems
The paper under review studies Liénard systems of the form \[ \dot x=y, \quad \dot y=-f_m(x)y-g_n(x) \] with the focus on the following two aspects: the existence of invariant algebraic curves and hyperelliptic limit cycles of the systems. The functions \(f_m(x)\) and \(g_n(x)\) involved are real polynomials of degree \(m\) and \(n\), respectively. One of the main results of the paper, Theorem 1.2, providing an estimation of the maximum number of hyperelliptic limit cycles, essentially improves Theorem 1.1, the counterpart of the paper [the first and the third author, ibid. 19, No. 1, Paper No. 43, 30 p. (2020; Zbl 1453.34045)]: More precisely, Theorem 1.2 dramatically narrows the gap between the upper and lower bounds of the maximum number of the limit cycles, which says as follows: If \(m+1\leq n \leq [\frac{4m+2}{3}]\), then any Liénard system of the above form can have at most \(n-m-1\) hyperelliptic limit cycles, and there does exist such a system which can reach this number. Notice that with the improvement of the estimation given in Theorem 1.2, together with the best estimations in the case (iii) (\(2m-1<n\leq 2m\)) and case (v) (\(n>2m+1\)) of Theorem 1.1, the authors in fact present a very satisfactory answer to the following special version of the 16th Hilbert problem: the upper bound of hyperelliptic limit cycles of the polynomial Liénard systems. In fact, the paper provides more than that, i.e., in all the cases, it gives lower bounds as well. Another interesting topic the paper considers is invariant algebraic curve. As it is well-known, hyperelliptic limit cycles, as a special type of algebraic limit cycles, whenever exist, must be contained in invariant algebraic curves of the system. Therefore the existence of such curves is an important indicator in considering the related subject. The case that \(n=2m+1\) is particularly difficult since it is extremely degenerate. Very few results are known, no systematic methods are available. To exhibit the difficulty and degeneracy of the case, the authors in Theorem 1.3 present an explicit example and demonstrate that such a system can admit infinite many invariant algebraic curves of arbitrary degrees. This example is quite amazing as perhaps it can reveal some deep relation between algebraic properties and the rich dynamic phenomenon of the problem. Finally, as an attempt to deepen the investigation of the degenerate case, the paper studies a concrete setting, \(m=2\), \(n=5\), where some very smart methods with certain geometric flavor are deliberately developed.
scientific article; zbMATH DE number 7365494