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Integrable and non-integrable Lotka-Volterra systems. (English) Zbl 07409878

Summary: In a recent paper [1], completely integrable cases were discovered of the Lotka Volterra Hamiltonian (LVH) system without linear terms, \(\frac{dx_i}{dt} = \dot{x}_i = \sum_{j = 1}^n a_{ij} x_i x_j\), \(i = 1, \dots, n\), \(a_{ij} = - a_{ji}\), satisfying the condition \(H = \sum_{i = 1}^n x_i = h = const\). In this paper, we first generalize this system to one that includes an arbitrary set of linear terms that preserve the Hamiltonian integral. We thus discover a wide class of LVH systems which we claim to be integrable, since their equations possess the Painlevé property, i.e. their solutions have only poles as movable singularities in the complex \(t\)-plane. Next, we focus on the case \(n = 3\) and vary some of the parameters, including additional nonlinearities to look for nonintegrable extensions with interesting dynamical properties. Our results suggest that, in this class of systems, non-integrability generally yields simple dynamics far removed from the type of complexity one expects from non-integrable 3-dimensional nonlinear systems.

MSC:

81-XX Quantum theory
82-XX Statistical mechanics, structure of matter
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