Symmetry, Singularities, and Integrability in Complex Dynamics II. Rescaling and Time-Translation in Two-Dimensional Systems

Abstract

The explicit integrability of second-order ordinary differential equations invariant under time-translation and rescaling is investigated. Quadratic systems generated from the linearisable version of this class of equations are analysed to determine the relationship between the Painlevé and singularity properties of the different systems. The transformation contains a parameter and for critical values, intimately related to the possession of the Painlevé property in the parent second-order equation, one finds a difference from the generic behaviour. This study is a prelude to a full discussion of the class of transformations which preserve the Painlevé property in the construction of quadratic systems from scalar nth-order ordinary differential equations invariant under time invariant under time translation and rescaling.

Renormalization Group as a Probe for Dynamical Systems

“…It has also been investigated as an oscillator of Liénard type by Chandrasekhar et al [189,269]. Over the past three decades or so the symmetry, solvability and integrability properties of this equation alone has been studied by several authors [282][283][284][285][286][287][288][289][290][291][292][293][294][295]. This equation and its variant are well known to physicists and mathematicians for over a century [277,278,296] because it is known to arise in various contexts.…”

Nonlinear ordinary differential equations: A discussion on symmetries and singularities

“…The discussion is mutatis mutandis identical.) However, this very convenient criterion has been shown, by way of counterexample, to be too strong [29].…”

Solution of Modified Equations of Emden-Type by Differential Transform Method

“…Physicists have found this equation in the study of equilibrium configurations of a spherical gas cloud acting under the mutual attraction of its molecules and subject to the laws of thermodynamics [9][10][11][12], in spherically symmetric expansion or collapse of a relativistic gravitating mass [13] and in the modeling of the fusion of pellets [14]. The invariance and the integrability of this equation have been a subject of study for the past two decades by a number of authors [15][16][17][18][19][20][21][22][23][24][25][26]. This equation have been found to possess an explicit general solution for the following parametric…”