2014/12/12 by Takafumi Mase, T. Mase, Ralph Willox +5 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Essential singularity #Gravitational singularity #Integrable system #Nonlinear Waves and Solitons #Numerical methods for differential equations #Regularization (linguistics) #Singularity #Singularity theory #Transformation (genetics) #math-ph #math.MP #nlin.SI
paper · pdf · doi:10.1098/rspa.2014.0956
19 pages, 10 figures
arxiv created 2014/12/12 · openalex publication_date 2015/11/01 · arxiv updated 2016/02/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
The ‘deautonomization’ of an integrable mapping of the plane consists in treating the free parameters in the mapping as functions of the independent variable, the precise expressions of which are to be determined with the help of a suitable criterion for integrability. Standard practice is to use the singularity confinement criterion and to require that singularities be confined at the very first opportunity. An algebro-geometrical analysis will show that confinement at a later stage leads to a non-integrable deautonomized system, thus justifying the standard singularity confinement approach. In particular, it will be shown on some selected examples of discrete Painlevé equations, how their regularization through blow-up yields exactly the same conditions on the parameters in the mapping as the singularity confinement criterion. Moreover, for all these examples, it will be shown that the conditions on the parameters are in fact equivalent to a linear transformation on part of the Picard group, obtained from the blow-up.