Singularity confinement and algebraic entropy: the case of the discrete Painlevé equations

Details Singularity confinement and algebraic entropy: the case of the discrete Painlevé equations Singularity confinement and algebraic entropy: the case of the discrete Painlevé equations

1999-10-18

arxiv.org/abs/solv-int/9910007v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

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Scientific paper

10.1016/S0375-9601(99)00670-2

We examine the validity of the results obtained with the singularity confinement integrability criterion in the case of discrete Painlev\'e equations. The method used is based on the requirement of non-exponential growth of the homogeneous degree of the iterate of the mapping. We show that when we start from an integrable autonomous mapping and deautonomise it using singularity confinement the degrees of growth of the nonautonomous mapping and of the autonomous one are identical. Thus this low-growth based approach is compatible with the integrability of the results obtained through singularity confinement. The origin of the singularity confinement property and its necessary character for integrability are also analysed.

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