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

1. 我们得到了几个新的具有前后不对称性的非自治非线性可积二阶常差分方程。通过适当的因变量变换，将这些前后向非对称可积映射转化为一类离散的painlevel方程。并将这些系统推广到奇偶不对称形式。揭示了连续Painleve方程在奇点周围的Laurent级数展开与离散Painleve方程奇点约束中的小参数之间的深刻关系。利用这两个小参数的共同特征，提出了一种求无穷小对称的方法，并应用于简单的算例。通过对可线性映射应用因变量变换，得到了具有正代数熵的非自治非线性可积二阶常差分方程。这个显著的例子给出了离散可积准则的一个新的方面。研究了一类三阶可积映射的初始条件迭代的次增长，这类映射是由离散Painleve方程耦合到一个同列映射得到的。我们已经证明了它的次数像n^3一样增长。在特殊情况下，映射满足奇点约束条件，我们发现一个较慢的二次增长。最后给出了一种构造级数增长为n^ n的可积n阶映射的方法。

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.