Centroaffine comitants of a quadratic-cubic two-dimensional differential system. (Russian) Zbl 0586.34042

Danilyuk, V. I.

Mat. Issled. 80, 57-66 (1985).

Consider the system (*) \(dx^ j/dt=a^ j_{\alpha \beta}x^{\alpha}x^{\beta}+a^ j_{\alpha \beta \gamma}x^{\alpha}x^{\beta}x^{\gamma}\) \((j,\alpha,\beta,\gamma =1,2)\), where \(a^ j_{\alpha \beta}\) and \(a^ j_{\alpha \beta \gamma}\) are symmetric tensors with respect to \(\alpha\),\(\beta\),\(\gamma\). A centroaffine comitant for (*) is of the type \((q,h_ 1,h_ 2)\), if there exists a homogeneous polynomial of degree \(q,h_ 1,h_ 2\) with respect to the coordinates of \(x^{\delta}\), \(a^ j_{\alpha \beta}\), \(a^ j_{\alpha \beta \gamma}\) respectively. The number \(h=h_ 1+h_ 2\) is called the order of such a comitant. It is known that the minimal polynomial basis of comitants for a homogeneous quadratic and for a homogeneous cubic system contains eight invariants and twenty comitants of order \(k\leq 4\). This paper is devoted to the construction of a minimal polynomial basis of comitants and of invariants for (*) up to fourth order.

Reviewer: P.Talpalaru

MSC:

37-XX

Dynamical systems and ergodic theory

Keywords:

symmetric tensors; centroaffine comitant; minimal polynomial basis