带双参数的第二类FKS函数方程的单谷延拓解

On Single-Valley-Extended Solution for the Second Type of FKS Equation with Two Parameters

FKS函数方程刻画了圆映射的拟周期到混沌的路径.作为简化,考虑了另外一种形式的方程——带双参数的第二类FKS函数方程,将单参数的情形拓展到双参数.首先分别给出了连续解和单谷延拓解满足的必要条件;其次,得到了单谷解和单谷延拓解的区分条件;然后,针对参数特定的取值范围,利用迭代构造法,分别构造了所有单谷解和单谷延拓解,将第二类KFS函数方程已有的递减解的结果扩充到单谷延拓连续解,并在文中给出了详细的证明过程.

The FKS function equation copes with the description of the quasiperiodic route to chaos for mappings of circle.As a simplification,this paper considers another form of equation which is the second type of FKS function equation with two parameters,extending the case of single parameter to two parameters.Firstly,the necessary conditions for continuous solutions and single-valley-extended solutions have been given respectively.Secondly,the conditions have been obtained for distinguishing single-valley solutions and single-valley-extended solutions.Finally,according to the specific value range of parameters,using the iterative construction method,all single-valley solutions and single-valley-extended solutions have been constructed,the result of the existing decreasing solution of the second type of KFS function equation is extended to the single-valley-extended continuous solution.The detailed proof process is given in this paper.