P***a
发帖数: 9 | 1
If one of A and B is 0, the answer is 0.
If both are positive, the limit of A and B can be proven to
exist, and they are equal.
It's easy to obtain the limit of A and B by programming,
given any specific A and B, because the interval of [B_n,
A_n] is non-increasing (A_n is non_increasing and B_n is
non-decreasing).
But I am not sure whether you have to be forced to get the
analytical solution of the limit of A and B? A pure
mathematical question?
I cannot give the analytical formula. But this q |
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s***e
发帖数: 911 | 2
This dynamics is:
x_{i+1}=(1/2)*(x_{i}+y_{i})
y_{i+1}=Sqrt[x_{i}*y_{i}]
This fixed points are determined:
x=(1/2)*(x+y)=>x=y
y=Sqrt[x*y]=y
也就是说任何二维点(x,x)都是该映射不动点. 在这不动点处分析稳定性:
映色Jacobi为:
A={{1/2,1/2},{1/2,1/2}}
特征值是: lambda=0, 和lambda=1.
lamda=0对应和45度线垂直的特征矢量; lambda=1对应和45度线平行的特征矢量.
也就是说, 45度线是该映射的稳定流形. 任何初始点映射足够长时间, 终点都在
这45度线上. 轨迹靠近直线处,和直线垂直.
粗推了一下,没仔细考虑.错漏别笑, hoho...
附录: 有关概念简介和这个推导的补充说明:
叠代动力学是指:
R_{i+1}=M(R_{i})
其中R是n维矢量(上诉例子就是二维的叠代), M是映射(动力学). 这个叠代M的不动
点R_{c}指这样的点:
M(R_{c})=R_{c}
这个方程也就 |
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