In mathematics, and in particular the study of dynamical 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 an attractor or repellor. In the case of hyperbolic dynamics, the corresponding notion is that of the hyperbolic set.
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 be a topological space, and a homeomorphism. If is a fixed point for , the stable set of is defined by
and the unstable set of is defined by
Here, denotes the inverse of the function , i.e.
, where is the identity map on .
If is a periodic point of least period , then it is a fixed point of , and the stable and unstable sets of are
Given a neighborhood of , the local stable and unstable sets of are defined by
If is metrizable, we can define the stable and unstable sets for any point by
where is a metric for . This definition clearly coincides with the previous one when is a periodic point.
Suppose now that is a compact smooth manifold, and is a diffeomorphism, . If is a hyperbolic periodic point, the stable manifold theorem assures that for some neighborhood of , the local stable and unstable sets are embedded disks, whose tangent spaces at are and (the stable and unstable spaces of ), respectively; moreover, they vary continuously (in a certain sense) in a neighborhood of in the topology of (the space of all diffeomorphisms from to itself). Finally, the stable and unstable sets are 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 some hyperbolic set (stable manifold theorem for hyperbolic sets).
If is a (finite-dimensional) vector space and an isomorphism, its stable and unstable sets are called stable space and unstable space, respectively.
- Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin/Cummings Publishing, Reading Mass. ISBN 0-8053-0102-X
- S. S. Sritharan, "Invariant Manifold Theory for Hydrodynamic Transition", (1990), John Wiley & Sons, NY, ISBN 0-582-06781-2
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