A class of planar \(n\) (even number)-polynomial system with a weak focus of order \(n^ 2-n\)
The following three interesting theorems are proved: Theorem 1. Let \(n=2m\), consider the system \[ {dx\over dt} =-y(1-x^{n- 1}-\alpha y^{n-1})+\sum_{k=m+1}^ n \varepsilon_ k[(2n- 2k+1)x^{2k-n} y^{2n-2k}+(n-2k)x^{2k-n-2} y^{2n-2k+2}],\tag{1} \] \[ {dy\over dt} = x(1-x^{n-1}-y^{n-1})+(n-1) \sum_{k=1}^ m \varepsilon_ k x^{2k-2} y^{n-2k+2} \] where \(\alpha\) is a transcendental number, \({\overset \rightharpoonup\varepsilon}=(\varepsilon_ 1,\varepsilon_ 2,\dots,\varepsilon_ n)\) is a small parameter. Then the origin of (1) is a weak focus of order \(n^ 2-n\), if \({\overset \rightharpoonup \varepsilon}\) is well chosen. Theorem 2. For any even number \(n\), there exists a class of \(n\)- polynomial systems with a weak focus of order \(n^ 2-n\), and \(n^ 2-n\) limit cycles can be generated in a neighborhood of this focus by small perturbation. Theorem 3. The origin of the system \[ {dx\over dt} = y(1-x^{n-1}-\beta y^{n-1})+\sum_{k=m+1}^ n \delta_ k[(2n-2k+1)x^{2k-n} y^{2n- 2k} +(2k-n) x^{2k-n-2} y^{2n-2k+2}], \] \[ {dy \over dt} = x(1-x^{n- 1} -\beta y^{n-1})+(n-1) \sum_{k=1}^ m \delta_ k x^{2k-2} y^{n-2k+2} \] is a saddle point of order \(n^ 2-n\), if \(\beta\) and \({\overset \rightharpoonup \delta}=(\delta_ 1,\delta_ 2,\dots,\delta_ n)\) are well chosen.
scientific article; zbMATH DE number 148464
scientific article