Limit cycles of polynomial differential systems bifurcating from the periodic orbits of a linear differential system in \(\mathbb R^d\)
The authors investigate the number of limit cycles of the following polynomial system \[ x_1' = -x_2 + \varepsilon P_1(x_1, \dots, x_d) + \varepsilon^2Q_1(x_1, \dots, x_d), \] \[ x_2' = x_1 + \varepsilon P_2(x_1, \dots, x_d) +\varepsilon^2Q_2(x_1, \dots, x_d), \] \[ x_k' = \varepsilon P_k(x_1, \ldots, x_d) + \varepsilon^2Q_k(x_1, \ldots, x_d) \] for \(k = 3,\dots, d\), where the maximum degree of \(P_k\) and \(Q_k\) is \(n_k\). When \(d \geq 2\) and \(\max\{n_1, n_2\}\geq 2\), by applying the first order averaging method to the above system, it is proved that at most \([(m-1)/2]n_3n_4\cdots n_d\) limit cycles bifurcate from the periodic orbits of the linear system with \(\varepsilon =0\). Further, by applying the second order averaging method to the above system, it is shown that at most \((m-1)(2n_3-1)(n_3+n_4-1)\cdots(n_3+n_d-1)\) limit cycles bifurcate from the periodic orbits of the linear system with \(\varepsilon = 0\). Although there are plenty of research publications on limit cycles of planar polynomial systems, not many have dealt with limit cycles of higher dimensional polynomial systems. This is an interesting paper.
scientific article; zbMATH DE number 5608810