On the integrability of some three-dimensional Lotka-Volterra equations with rank-$1$ resonances
We investigate the local integrability in C3 of some three-dimensional Lotka–Volterra equations at the origin with (p : q : r)-resonance, x˙ = P = x(p + ax + by + cz), y˙ = Q = y(q + dx + ey + fz), z˙ = R = z(r + gx + hy + kz). Recent work on this problem has centered on the case where the resonance is of “rank-2”. That is, there are two independent linear dependencies of p, q and r over Q. Here, we consider some situations where there is only one such dependency. In particular, we give necessary and sufficient conditions for integrability for the case of (i, −i, λ)-resonance with λ /∈ iR (after a scaling, this is just the case p + q = 0 with q/r /∈ R), and also the case of (i − 1, −i − 1, 2)-resonance (a subcase of p + q + r = 0) under the additional assumption that a = k = 0. Our necessary and sufficient conditions for integrability are given via the search for two independent first integrals of the form x αy βz γ(1 + O(x, y, z). However, a new feature in the case of rank-1 resonance is that there is a distinguished choice of analytic first integral, and hence it makes sense to seek conditions for just one (analytic) first integral to exist. We give necessary and sufficient conditions for just one first integral for the two families of systems mentioned above, except that for the second family some of the cases of sufficiency have been left as conjectural.
Normalizability, integrability and monodromy maps of singularities in three-dimensional vector fields
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 and 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
Normed units in abelian group rings
We compute explicitly the isomorphic structure of the normalized unit group of an abelian group ring under some minimal natural restrictions on the group basis and the coefficient ring. This enlarges affirmations due to Chatzidakis-Pappas (J. London Math. Soc., 1991) and Mollov (Publ. Math. Debrecen, 1971).