Title:

Normalizability, integrability and monodromy maps of singularities in threedimensional vector fields

In this thesis we consider threedimensional dynamical systems in the neighbourhood of a singular point with rankone and ranktwo resonant eigenvalues. We first introduce and generalize here a new technique extending previous work which was described by Aziz an Christopher (2012), where a second first integral of a 3D system can be found if the system has a Darbouxanalytic first integral and an inverse Jacobi multiplier. We use this new technique to find two independent first integrals one of which contains logarithmic terms, allowing for nonzero resonant terms in the formal normal form of vector field. We also consider sufficient conditions for the existence of one analytic first integral for three dimensional vector fields around a singularity. Starting from the generalized LotkaVolterra system with rankone resonant eigenvalues, using the normal form method, we find an inverse Jacobi multiplier of the system under suitable conditions. Moreover, these conditions are sufficient conditions for the existence of one analytic first integral of the system. We apply this to demonstrate the sufficiency of the conditions in Aziz and Christopher (2014). In the case of twodimensional systems, Christopher et al (2003) addressed the question of orbital normalizability, integrability, normalizability and linearizability of a complex differential system in the neighbourhood at a critical point. We here address the question of normalizability, orbital normalizability, and integrability of threedimensional systems in the neighbourhood at the origin for rankone resonance system. We consider the case when the eigenvalues of threedimensional systems have rankone resonance satisfying the condition the sum of eigenvalues is equal to zero a typical example, and we use a further change of coordinates to bring the formal normal form for threedimensional systems into a reduced normal form which contains a finite number of resonant monomials. By using this technique, we can find two independent first integrals formally. The first one of these first integrals is of Darbouxanalytic type, and other first integral contains logarithmic terms corresponding to nonzero resonant monomials of the original system. We introduce the monodromy map in threedimensional vector fields by using these two independent first integrals to study a relationship between normalizability and integrability of systems. In the case of rankone resonant eigenvalues, we get a monodromy map which is in normal form, and then in the same way as the case of vector fields, we use a further change of coordinates to reduce this map into a reduced map which contains only a finite number of resonant monomials. This thesis also examines briefly the case of ranktwo resonant eigenvalues of threedimensional systems. The normal form in this case contains an infinite number of resonant monomials, we were not able to find a reduced normal form with a finite number of resonant monomials. This situation is therefore much more complex than the rankone case. Thus, we simplify the investigation by truncating the 3D system to a 3D homogeneous cubic system as a first step to understanding the general case. Even though we can find two independent first integrals, the second one involves the hypergeometric function, leading to some interesting topics for further investigation.
