On generalized Volterra systems

Abstract

a b s t r a c tWe construct a large family of evidently integrable Hamiltonian systems which are generalizations of the KM system. The algorithm uses the root system of a complex simple Lie algebra. The Hamiltonian vector field is homogeneous cubic but in a number of cases a simple change of variables transforms such a system to a quadratic Lotka-Volterra system. We present in detail all such systems in the cases of A 3 , A 4 and we also give some examples from higher dimensions. We classify all possible Lotka-Volterra systems that arise via this algorithm in the A n case.

Integrable reductions of the Bogoyavlenskij-Itoh Lotka-Volterra systems

“…This proves the different claims in (4). We next prove (5). In view of items (1) and (4) we need to show that…”

“…By now, many systems of the form (1.1) have been introduced and studied, often from the point of (Liouville, Darboux or algebraic) integrability [2,3,10,19,17,9,13,6,4] or Lie theory [2,3,7,1,5], but also in relation with other integrable systems [18,8].…”

Congruence method for global Darboux reduction of finite-dimensional Poisson systems

“…Recall that Darboux' theorem is not constructive (it is an existence theorem) and it is only local in scope, being valid in principle in a neighborhood of every point x. In practice, such reduction has been constructed globally for many solution families of structure matrices [2]- [9], while in many other cases there is no known global reduction (which, in fact, might not exist). In a similar way, the existence of a congruence reduction is not guaranteed in advance (in other words, a congruence matrix always exists but it is not necessarily a Jacobian matrix, as shown in Theorem 1).…”

“…In spite that the Darboux canonical form always exists, Darboux theorem is not constructive (it is an existence theorem) and applies only locally (in the neigborhood of every point, in domains in which the structure matrix has constant rank). However, diverse efforts have been devoted to the explicit and global construction of the Darboux coordinates (for instance, see [2]- [9]). This is relevant not only as a natural way to improve the scope of Darboux' theorem, but also as a useful reduction of a Poisson system into a classical Hamiltonian flow (often of lower dimension) with the advantages and plethora of specific methods inherent to the latter format.…”

On the liouville intergrability of Lotka-Volterra systems

“…In Charalambides et al [27] the algorithm was generalized as follows. Consider a subset of + such that ⊂ ⊂ + .…”