Properties of the Dominant Behaviour of Quadratic Systems

Abstract

We study the dominant terms of systems of Lotka-Volterra-type which arise in the the mathematical modelling of the evolution of many divers natural systems from the viewpoint of both symmetry and singularity analyses. The connections between an increase in the amount of symmetry possessed by the system and the possession of the Painlevé Property are noted. For specific values of the parameters of the system we see that possession of the Painlevé Property is characterised by a Left Painlevé Series rather than the more standard Right Painlevé Series.

Integrability analysis of the Emden-Fowler equation

“…This does reinforce the close relationship between the Lie and Painlevé analyses observed previously [28,45,49,5,6]. In the case of rational n, it was shown that, for specific values of n in the range (1, quadrature (2.47) cannot, as yet, be evaluated for these values, noting the results in [60] and [10] we believe that the evaluation thereof is only a matter of time and effort.…”

Linearisation of a second-order nonlinear ordinary differential equation

“…In the search for solutions of differential equations, one discovers the beauty of the algebraic properties that the equations possess. Even though closed-form solutions are the primary objective, one cannot ignore the interesting properties of the equations [1][2][3][4][5][6]. In recent years, one such area in relativistic astrophysics involves the embedding of a four-dimensional differentiable manifold into a higher dimensional Euclidean space which gives rise to the so-called Karmarkar condition for Class I spacetimes [7].…”

Nonlinear juvenile predation population dynamics

“…Paul Painlevé took these ideas and classified ordinary differential equations (ODEs) of second order according to the type of singularities of their solutions [8]. Since then, the property has been used to construct symmetries, to find explicit solutions, to detect control parameters, and so on [9,10]. Basically, a system of ODEs has the Painlevé property if its general solution has no movable critical singular points.…”