|
|
| Line 1: |
Line 1: |
| 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]].
| | Wilber Berryhill is what his spouse loves to contact him and he totally enjoys this name. She is really fond of caving but she doesn't have the time lately. Her family members lives in Alaska but her spouse desires them to transfer. For years she's been operating as a journey agent.<br><br>Have a look at my web page: [http://kjhkkb.net/xe/notice/374835 online psychic chat] |
| | |
| ==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 (mathematics)|flow]].
| |
| | |
| Let <math>X</math> be a [[topological space]], and <math>f\colon X\to X</math> a [[homeomorphism]]. If <math>p</math> is a [[Fixed point (mathematics)|fixed point]] for <math>f</math>, the '''stable set of <math>p</math>''' is defined by
| |
| | |
| :<math>W^s(f,p) =\{q\in X: f^n(q)\to p \mbox{ as } n\to \infty \}</math>
| |
| and the '''unstable set of <math>p</math>''' is defined by
| |
| :<math>W^u(f,p) =\{q\in X: f^{-n}(q)\to p \mbox{ as } n\to \infty \}.</math>
| |
| | |
| Here, <math>f^{-1}</math> denotes the [[Inverse function|inverse]] of the function <math>f</math>, i.e.
| |
| <math>f\circ f^{-1}=f^{-1}\circ f =id_{X}</math>, where <math>id_{X}</math> is the identity map on <math>X</math>.
| |
| | |
| If <math>p</math> is a [[periodic point]] of least period <math>k</math>, then it is a fixed point of <math>f^k</math>, and the stable and unstable sets of <math>p</math> are
| |
| | |
| :<math>W^s(f,p) = W^s(f^k,p)</math> | |
| and
| |
| :<math>W^u(f,p) = W^u(f^k,p).</math>
| |
| | |
| Given a [[neighborhood (mathematics)|neighborhood]] <math>U</math> of <math>p</math>, the '''local stable and unstable sets''' of <math>p</math> are defined by
| |
| | |
| :<math>W^s_{\mathrm{loc}}(f,p,U) = \{q\in U: f^n(q)\in U \mbox{ for each } n\geq 0\} </math> | |
| and
| |
| :<math>W^u_{\mathrm{loc}}(f,p,U) = W^s_{\mathrm{loc}}(f^{-1},p,U).</math>
| |
| | |
| If <math>X</math> is [[metrizable]], we can define the stable and unstable sets for any point by
| |
| | |
| :<math>W^s(f,p) = \{q\in X: d(f^n(q),f^n(p))\to 0 \mbox { for } n\to \infty \}</math>
| |
| and
| |
| :<math>W^u(f,p) = W^s(f^{-1},p),</math>
| |
| | |
| where <math>d</math> is a [[Metric (mathematics)|metric]] for <math>X</math>. This definition clearly coincides with the previous one when <math>p</math> is a periodic point.
| |
| | |
| Suppose now that <math>X</math> is a [[compact space|compact]] [[smooth manifold]], and <math>f</math> is a <math>\mathcal{C}^k</math> [[diffeomorphism]], <math>k\geq 1</math>. If <math>p</math> is a hyperbolic periodic point, the [[stable manifold theorem]] assures that for some neighborhood <math>U</math> of <math>p</math>, the local stable and unstable sets are <math>\mathcal{C}^k</math> embedded disks, whose [[tangent space]]s at <math>p</math> are <math>E^s</math> and <math>E^u</math> (the stable and unstable spaces of <math>Df(p)</math>), respectively; moreover, they vary continuously (in a certain sense) in a neighborhood of <math>f</math> in the <math>\mathcal{C}^k</math> topology of <math>\mathrm{Diff}^k(X)</math> (the space of all <math>\mathcal{C}^k</math> diffeomorphisms from <math>X</math> to itself). Finally, the stable and unstable sets are <math>\mathcal{C}^k</math> 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).
| |
| | |
| ==Remark==
| |
| If <math>X</math> is a (finite dimensional) vector space and <math>f</math> an isomorphism, its stable and unstable sets are called stable space and unstable space, respectively.
| |
| | |
| ==See 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
| |
| | |
| * S. S. Sritharan, "Invariant Manifold Theory for Hydrodynamic Transition", (1990), John Wiley & Sons, NY, ISBN 0-582-06781-2
| |
| ISBN 978-0-582-06781-3
| |
| | |
| {{PlanetMath attribution|id=4357|title=Stable manifold}}
| |
| | |
| [[Category:Limit sets]]
| |
| [[Category:Dynamical systems]]
| |
| [[Category:Manifolds]]
| |
Wilber Berryhill is what his spouse loves to contact him and he totally enjoys this name. She is really fond of caving but she doesn't have the time lately. Her family members lives in Alaska but her spouse desires them to transfer. For years she's been operating as a journey agent.
Have a look at my web page: online psychic chat