Dynamics of normalized systems on surfaces (Q1769843)

This paper is an extension of previous results obtained by the author on the dynamics of commuting systems on two-dimensional manifolds [Ann. Mat. Pura Appl., IV. Ser. 173, 213--232 (1997; Zbl 0941.34018)]. More precisely, it is on a pair of two-dimensional autonomous ordinary differential equations satisfying some conditions for commuting. Among the earlier results can be noted: centers are isochronous, critical (or stationary) points have positive index, limit cycles do not exist, attraction and central regions are unbounded. The notions of normalizer and commutators are introduced. It is claimed that passing from commutators to normalizers one has the advantage to consider a wider class of systems. In this framework, several dynamical properties of couples of commuting systems are extended to normalized systems. A coarse classification of the dynamics generated by normalized vector fields on two-dimensional compact connected oriented manifolds is presented.