Symmetry, singularities and integrability in complex dynamics II: Rescaling and time-translation in two-dimensional systems
نویسندگان
چکیده
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 translation and rescaling.
منابع مشابه
Symmetry, singularities and intregrability in complex dynamics VII: Integrability Properties of FRW-Scalar Cosmologies
This paper considers some physically interesting cosmological dynamical systems in the FRW-scalarfield category which are examined for integrability according to the criterion of Painlevé. In the literature these systems have been examined from the point of view of dynamical systems and the results from the two disparate methods of analysis are compared. This allows some more general comments t...
متن کاملSymmetry, singularities and integrability in complex dynamics V: Complete symmetry groups of certain relativistic spherically symmetric systems
Abstract: We show that the concept of complete symmetry group introduced by Krause (J Math Phys 35 (1994) 5734-5748) in the context of the Newtonian Kepler problem has wider applicability, extending to the relativistic context of the Einstein equations describing spherically symmetric bodies with certain conformal Killing symmetries. We also provide a simple demonstration of the nonuniqueness o...
متن کاملThe Paradox of Intervening in Complex Adaptive Systems; Comment on “Using Complexity and Network Concepts to Inform Healthcare Knowledge Translation”
This commentary addresses two points raised by Kitson and colleagues’ article. First, increasing interest in applying the Complexity Theory lens in healthcare needs further systematic work to create some commonality between concepts used. Second, our need to adopt a better understanding of how these systems organise so we can change the systems overall behaviour, creates a paradox. We seek to m...
متن کاملSymmetry, Singularities and Integrability in Complex Dynamics I: The Reduction Problem
Quadratic systems generated using Yang-Baxter equations are integrable in a sense, but we display a deterioration in the possession of the Painlevé property as the number of equations in each ‘integrable system’ increases. Certain intermediate systems are constructed and also tested for the Painlevé property. The Lie symmetries are also computed for completeness.
متن کاملSymmetry, Singularities and Integrability in Complex Dynamics Iv: Painlevé Integrability of Isotropic Cosmologies
In the modelling of various phenomena in relativistic cosmology the final result of the modelling process is a system of ordinary differential equations, whose solution is by no means obvious. Generally these systems are of the first order and autonomous, so that, if the system is two-dimensional, one is assured of the existence of solutions under mild conditions on the terms in the equations. ...
متن کامل