Study on Construction and Classification of Nonautonomous Nonlinear Integrable Systems Based on Symmetry of Bilinear Form

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1. We found several new nonautonomous nonlinear integrable second order ordinary difference equations which have forward-backward asymmetry. By using suitable dependent variable transformations, those forward-backward asymmetric integrable mappings are transformed into a class of discrete Painleve equations. Moreover we generalized those systems to even-odd asymmetric forms.2. We revealed the deep relation between the Laurent series expansion around a singular point for continuous Painleve equations and the small parameter in singularity confinement for discrete Painleve equations. By using the common feature for those two small parameters, we formulated a method to find an infinitesimal symmetry and applied it for simple examples.3. It was shown that by applying some dependent variable transformations for linearizable mappings we can create nonautonomous nonlinear integrable second order ordinary difference equations which have positive algebraic entropy. This remarkable example give a new aspect for the discrete integrability criteria.4. We studies the degree growth of the iterates of the initial conditions for a class of third-order integrable mappings which result from the coupling of a discrete Painleve equation to an homographic mapping. We showed that the degree grows like n^3. In the special cases where the mapping satisfies the singularity confinement requirement we find a slower, quadratic growth. Finally we presented a method for the construction of integrable Nth-order mapping with degree growth n^N.