- Stable manifold
In
mathematics , and in particular the study ofdynamical systems , the idea of "stable and unstable sets" or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of anattractor orrepellor . In the case ofhyperbolic dynamics , the corresponding notion is that of thehyperbolic set .Definition
The following provides a definition for the case of a system that is either an
iterated function or has discrete-time dynamics. Similar notions apply for systems whose time evolution is given by a flow.Let X be a
topological space , and fcolon X o X ahomeomorphism . If p is afixed point for f, the stable set of p is defined by:W^s(f,p) ={qin X: f^n(q) o p mbox{ as } n o infty }and the unstable set of p is defined by:W^u(f,p) ={qin X: f^{-n}(q) o p mbox{ as } n o infty }.
Here, f^{-1} denotes the
inverse of the function f, i.e.fcirc f^{-1}=f^{-1}circ f =id_{X}, where id_{X} is the identity map on X.If p is a
periodic point of least period k, then it is a fixed point of f^k, and the stable and unstable sets of p are:W^s(f,p) = W^s(f^k,p)and:W^u(f,p) = W^u(f^k,p).
Given a neighborhood U of p, the local stable and unstable sets of p are defined by
:W^s_{mathrm{loc(f,p,U) = {qin U: f^n(q)in U mbox{ for each } ngeq 0} and:W^u_{mathrm{loc(f,p,U) = W^s_{mathrm{loc(f^{-1},p,U).
If X is
metrizable , we can define the stable and unstable sets for any point by:W^s(f,p) = {qin X: d(f^n(q),f^n(p)) o 0 mbox { for } n o infty }and:W^u(f,p) = W^s(f^{-1},p),
where d is a metric for X. This definition clearly coincides with the previous one when p is a periodic point.
Suppose now that X is a
compact smooth manifold , and f is a mathcal{C}^kdiffeomorphism , kgeq 1. If p is a hyperbolic periodic point, thestable manifold theorem assures that for some neighborhood U of p, the local stable and unstable sets are mathcal{C}^k embedded disks, whosetangent space s at p are E^s and E^u (the stable and unstable spaces of Df(p)), respectively; moreover, they vary continuously (in a certain sense) in a neighborhood of f in the mathcal{C}^k topology of mathrm{Diff}^k(X) (the space of all mathcal{C}^k diffeomorphisms from X to itself). Finally, the stable and unstable sets are mathcal{C}^k injectively immersed disks. This is why they are commonly called stable and unstable manifolds. This result is also valid for nonperiodic points, as long as they lie in somehyperbolic set (stable manifold theorem for hyperbolic sets).Remark
If X is a (finite dimensional) vector space and f an isomorphism, its stable and unstable sets are called
stable space andunstable space , respectively.ee also
*
Limit set
*Julia set
*Center manifold References
* Ralph Abraham and Jerrold E. Marsden, "Foundations of Mechanics", (1978) Benjamin/Cummings Publishing, Reading Mass. ISBN 0-8053-0102-X
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