Local invariant sets of analytic vector fields
- Lokal invariante Mengen analytischer Vektorfelder
Kruff, Niclas; Walcher, Sebastian (Thesis advisor); Zerz, Eva Barbara (Thesis advisor)
Dissertation / PhD Thesis
Dissertation, RWTH Aachen University, 2018
In this doctoral thesis we discuss invariant sets of autonomous ordinary differential equations. Finding and studying invariant sets is a common tool in order to examine stability properties of dynamical systems near stationary points or in general to analyse the local dynamics of ODEs. Particular invariant sets are for example given by algebraic invariant curves of plane polynomial vector fields and these play an important role in the question of integrability of a system and moreover in the study of the center problem. We first introduce the notion of semi-invariants and invariant ideals whose vanishing sets are local invariant sets of the corresponding system of ordinary differential equations. An essential tool for the study of the local behaviour near stationary points is the Poincaré-Dulac normal form (PDNF) of vector fields. We prove that every invariant ideal of a vector field in PDNF already is an invariant ideal of its semi-simple linear part. Moreover, we show that all invariant ideals can be generated by semi-invariants. Therefore, semi-invariants can be seen as building blocks of all invariant ideals. In order to prove these results, we use properties of the ring of formal power series. In general, finding degree bounds of semi-invariants for polynomial vector fields is a difficult task and it is still an open problem: the so-called Poincaré problem. One major objective of this thesis is to develop an approach to get degree bounds under certain genericity assumptions. We generalize some results in dimension two to higher dimensions which reveals a connection between the highest degree of the vector field and the highest degree of possible irreducible semi-invariants. The crucial technique that is involved to obtain theses degree bounds is the concept of stationary points at infinity. Moreover, invariant sets are useful in bifurcation theory. In this thesis we especially concentrate on Hopf bifurcations which lead to limit cycles which in turn correspond to periodic solutions of the differential equation. We provide a new algorithmic approach to determine the nature of the bifurcation. This algorithm allows us to compute the crucial quantities of a Hopf-bifurcation, particularly Lyapunov's first quantity, which allows to decide whether the occuring limit cycle is attracting or repelling. Finally, we tackle the nverse problem. Instead of finding invariant sets for a given vector field, we determine vector fields for a prescribed ideal which leave the vanishing set of this ideal invariant. We define attracting invariant varieties and investigate the particular situation of smooth and compact real varieties. For those varieties we construct a set of vector fields such that there exists a neighborhood of in which all solution trajectories are attracted by the variety.