Normalizability, integrability and monodromy maps of singularities in three-dimensional vector fields
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In this thesis we consider three-dimensional dynamical systems in the neighbourhood of a singular point with rank-one and rank-two 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 Darboux-analytic 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 non-zero 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 Lotka-Volterra system with rank-one 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 two-dimensional 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 three-dimensional systems in the neighbourhood at the origin for rank-one resonance system.
We consider the case when the eigenvalues of three-dimensional systems have rank-one 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 three-dimensional 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 Darboux-analytic type, and other first integral contains logarithmic terms corresponding to non-zero resonant monomials of the original system.
We introduce the monodromy map in three-dimensional vector fields by using these two independent first integrals to study a relationship between normalizability and integrability of systems. In the case of rank-one 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 rank-two resonant eigenvalues of three-dimensional 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 rank-one 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.
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