c**********w
发帖数: 1746 | 1 请原谅简单问题，我刚开始看dynamic systems，正在看saddle fixed point。这个地
方讲到，discrete-time system的unstable invariant manifolds和stable invariant
manifolds可以intersect at nonzero angle，但是continuous-time就不行。我完全
没搞懂，这个intersect是指相互无限接近类似limit point，还是指有非空交集? 为什
么discrete-time可以有transversal intersection, 而连续系统不行？区别在哪里？
刚刚学，问题很弱，谢谢指教！ | l********e
发帖数: 3632 | 2 你应该先去找离散系统transversality的定义啊。
invariant
【在 c**********w 的大作中提到】
: 请原谅简单问题，我刚开始看dynamic systems，正在看saddle fixed point。这个地
: 方讲到，discrete-time system的unstable invariant manifolds和stable invariant
: manifolds可以intersect at nonzero angle，但是continuous-time就不行。我完全
: 没搞懂，这个intersect是指相互无限接近类似limit point，还是指有非空交集? 为什
: 么discrete-time可以有transversal intersection, 而连续系统不行？区别在哪里？
: 刚刚学，问题很弱，谢谢指教！
| E*****T
发帖数: 1193 | 3 建议你换本书参考参考，可能你现在看的书太抽象？因为torus作为非一维的最简单例
子，就直观告诉你stable和unstable manifold怎么交了。
我知道的可能比你还皮毛，看的是Robert Devaney的那本书。 | z***c
发帖数: 102 | 4 稳定和不稳定流形的交集是个不变集。因为连续系统的轨道是光滑曲线，所以连续系统
稳定和不稳定流形至少会交于一条光滑曲线。从这个意义上说，连续系统的稳定和不稳
定流形不可能完全正交。不过这不是个大问题，只需要改变正交的定义，除去向量场的
方向就可以了。 | s*****e
发帖数: 115 | 5
This is not true in general!
None of you guys get it straight. The correct statement is that
Given a continuous-time dynamical system, the stable and unstable manifolds
of the SAME fixed point can never intersect transversally at any location
other than the fixed point itself. The meaning of "transversal intersection"
is exactly the same as its usual meaning in differential geometry.
For a continuous-time dynamical system (dimension 3 or higher), there can be
transversal intersections, for example, between stable and unstable
manifolds of two different fixed points (i.e., a heteroclinic connection),
or between stable and unstable manifolds of a periodic orbit, etc.
不过这不是个大问题，只需要改变正交的定义，除去向量场的
【在 z***c 的大作中提到】
: 稳定和不稳定流形的交集是个不变集。因为连续系统的轨道是光滑曲线，所以连续系统
: 稳定和不稳定流形至少会交于一条光滑曲线。从这个意义上说，连续系统的稳定和不稳
: 定流形不可能完全正交。不过这不是个大问题，只需要改变正交的定义，除去向量场的
: 方向就可以了。
| z***c
发帖数: 102 | 6 OK I was not being precise. But I think the OP was asking about the same
fixed point.
The concept of transversality in differential topology is the following: Two
submanifolds M, N of the manifold W are transversal at x if
TxM + TxN = TxW
the sum is in the sense of vector subspace. This is somewhat different from
the naive sense of transversality. If two submanifolds intersects at at a
point, then the transversality condition is the same as having "nonzero
angle".
As I explained, the intersection between stable and unstable manifolds must
have at least dimension 1. For the same fixed point,
dim Stable + dim Unstable = n = dim Whole space.
If stable and unstable intersects at one dimension, then
dim (TxStable + TxUnstable) <= n-1,
so the intersection can never be transversal.
For different fixed points or stable/unstable manifolds of normally
hyperbolic manifolds, it is possible that
dim (TxStable + TxUnstable) > n,
and transversal intersection is possible. |
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