Local invariant sets of analytic vector fields [ Back ]

UAB - Dept. Matemàtiques (C1/-128)
Niclas Kruff
Aachen University


In the theory of autonomous ordinary differential equations invariant sets play an important role. In particular, we are interested in local analytic invariant sets near stationary points. Invariant sets of a differential equation correspond to invariant ideals of the associated derivation in the power series algebra. Poincaré-Dulac normal forms are very useful in studying semi-invariants and invariant ideals. We prove that an invariant ideal with respect to a vector field, given in normal form, is already invariant with respect to the semi-simple part of its Jacobian at the stationary point. This generalizes a known result about semi-invariants, that is invariantsets of codimension 1. Moreover, we give a characterization of all ideals which areinvariant with respect to the semi-simple part of the Jacobian. As an application, we consider polynomial systems and we provide a sharp bound of the total degree of possible polynomial semi-invariants under some generic conditions