
	Where results make sense

Topic: Stable manifold

Related Topics

Riemannian manifold

complex manifold

differentiable manifolds

pseudo Riemannian manifold

smooth manifolds

symplectic manifolds

Calabi Yau manifolds

Variable Length Intake Manifold

topological manifolds

Flag manifold

Curvature of Riemannian manifolds

Poisson manifold

In the News (Fri 13 Nov 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	PlanetMath: stable manifold
		This result is also valid for nonperiodic points, as long as they lie in some hyperbolic set (stable manifold theorem for hyperbolic sets).
		Cross-references: hyperbolic set, manifolds, unstable, tangent spaces, stable manifold theorem, hyperbolic periodic point, diffeomorphism, smooth manifold, compact, metric, metrizable, neighborhood, least period, periodic point, stable, fixed point, homeomorphism, topological space
		This is version 7 of stable manifold, born on 2003-06-13, modified 2003-06-16.
	www.planetmath.org /encyclopedia/UnstableManifold.html (195 words)

	Multifario, Computing Invariant Manifolds (Site not responding. Last check: 2007-10-14)
		I wanted to include it because it indicates that the representation of a manifold as a set of overlapping charts may be used in situations other than implicitly defined manifolds, as long as there is a way to project and compute tangent spaces.
		The stable manifold of the origin as the maximum time changes.
		The unstable manifold of one of the fixedpoints at z=28.
	www.coin-or.org /multifario/Lorenz.html (310 words)

	Arneodo's system (Site not responding. Last check: 2007-10-14)
		In fact, also the stable manifold of this periodic orbit is topologically a Möbius strip.
		The two-dimensional unstable manifold of the equilibrium (
		The unstable manifold of B forms a generic heteroclinic intersection with the non-orientable stable manifold of the periodic orbit.
	www.maths.ex.ac.uk /~hinke/nonorientable (259 words)

	The Stable Manifold Theorem For Nonlinear Stochastic Systems With Memory Ii: The Local Stable Manifold Theorem. - ... (Site not responding. Last check: 2007-10-14)
		The stable and unstable manifolds are stationary and asymptotically invariant under the stochastic...
		1.2: The Stable Manifold Theorem For Nonlinear Stochastic..
		4 The stable manifold theorem for stochastic dierential equat..
	citeseer.ist.psu.edu /mohammed99stable.html (563 words)

	Globalizing 2D unstable manifolds of maps
		1994] the stable and unstable manifolds are qualitatively sketched in the
		Hence, one branch of the unstable manifold of the circle converges to the invariant circle
		The branches of the stable manifold are symptotic to the
	www.geom.uiuc.edu /docs/research/manifolds/hopfhopf.html (347 words)

	The Stable Manifold Theorem For Nonlinear Stochastic Systems With Memory - II: The Local Stable Manifold Theorem. - ... (Site not responding. Last check: 2007-10-14)
		27.8%: The Stable Manifold Theorem For Nonlinear Stochastic..
		6 The stable manifold theorem for non-linear stochastic heredi..
		2 The stable manifold theorem for stochastic dierential equat..
	citeseer.ist.psu.edu /280139.html (528 words)

	RECENT PUBLICATIONS - Salah-Eldin A. Mohammed (Site not responding. Last check: 2007-10-14)
		The stable and unstable manifolds are stationary, live in a stationary tubular neighborhood of the stationary solution and are asymptotically invariant under the stochastic semiflow of the see/spde.
		`` The Stable Manifold Theorem for Semilinear SPDEs" (with Tusheng Zhang and Huaizhong Zhao), (2003) pp.
		The unstable and stable manifolds are stationary, asymptotically invariant under the stochastic semiflow and have fixed finite dimension and codimension, respectively.
	salah.math.siu.edu /recentpub.html (2345 words)

	Accessible and Inaccessible Basin Boundary Points
		In particular, we show the following: (i) All basin boundary points that are accessible from the red region lie on the stable manifold of the fixed point (1/3,0) (and similarly for the blue region and the fixed point (0,0), and for the green region and the fixed point (2/3,0)).
		(iii) The stable manifolds of the fixed points (0,0), (1/3,0), (2/3,0) are each dense in the basin boundary (i.e., every neighborhood of a basin boundary point contains part of the stable manifolds of all three fixed points, and the basin boundary is thus the closure of the stable manifold of any of the fixed points).
		These stable manifold segments are Wada (they are part of the trapping region boundaries) and hence possess red, blue, and green striped regions accumulating on them.
	www-chaos.umd.edu /publications/wadabasin/node5.html (1554 words)

	Linear 2D systems
		The stable (unstable) manifold for a fixed point is the set of all points in the plane which tend to the fixed point as time goes to positive (negative) infinity.
		This is a one-dimensional set unlike the case of a node or vortex where it is either the whole plane (two-dimensional) or a single point ("zero" dimensional.) Similarly, the stable manifold for a saddle point is the eigenvector corresponding to the negative eigenvalue and is also one-dimensional.
		Except for degenerate cases, the sum of the dimensions of the unstable and stable manifolds is equal to 2, the dimension of the plane.
	www.cnbc.cmu.edu /~bard/xppfast/lin2d.html (829 words)

	Globalizing 2D unstable manifolds of maps
		Stable and unstable manifolds of invariant manifolds of saddle-type, here fixed points or invariant circles, play important roles in organizing the global dynamics.
		of an invariant manifold of saddle-type of a three-dimensional diffeomorphism.
		in a neighborhood of the invariant manifold of saddle-type.
	www.geom.uiuc.edu /docs/research/manifolds/intro.html (1007 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		Therefore the relevant quantities that we shall consider: tori, stable and unstable manifolds are all relative to the map.
		\eqno(1.2) $$ In analogy with the differential equations (0.0) we define: ${\Cal Y}^-_\omega=\{r^o\in B^2_\sigma\times{\Bbb R}\times{\Bbb T}\vert dist(\hat\phi_{2\pi n}^\mu(r^o),{\Cal T}_\omega)\to0$ as $n\to+\infty$ (the {\it stable manifold} at ${\Cal T}_\omega$) and ${\Cal Y}^+_\omega=\{r^o\in B^2_\sigma\times{\Bbb R}\times{\Bbb T}\vert dist(\hat\phi_{-2\pi n}^\mu(r^o),{\Cal T}_\omega)\to0$ as $n\to\infty$ (the {\it unstable manifold} at ${\Cal T}_\omega$).
		What follows is closely related to the Hadamard's theorem about the existence of stable and unstable manifolds of fully hyperbolic systems and in general to the theory of invariant sets for flows and maps defined on a manifold (possibly ${\Bbb R}^N$).
	www.ma.utexas.edu /mp_arc/papers/97-478 (1421 words)

	Globalizing 2D unstable manifolds of maps
		For this purpose we compute one branch of the unstable manifold of the circle of saddle-type that is far away from the attractor.
		Figure 11 shows that the unstable manifold converges to the attracting period-two circles by folding; the fold is clearly seen in the enlargement of Figure 11.
		We computed the stable and unstable manifolds that form the heteroclinic tangle, but we did not include the other stable and unstable manifolds in the figures.
	www.geom.uiuc.edu /docs/research/manifolds/henon.html (482 words)

	VUTH 98-21
		This circle is shown in green, its stable manifold is blue, and its unstable manifold is red.
		The stable manifold forms the boundary of the basin of attraction (114K).
		Note: In the paper Globalizing two-dimensional unstable manifolds of maps, we computed heteroclinic intersections of stable and unstable manifolds (for A = 0.1, b = 0.68 and c = 0.1) with the general 2D algorithm.
	www.enm.bris.ac.uk /staff/hinke/vuth98-21/index.html (390 words)

	The center manifold theorem
		When a system loses stability, the number of eigenvalues and eigenvectors which are associated with this change is typically small.
		Moreover, the stability of such a solution is determined by its stability within the center manifold.
		The formal Taylor series of the center manifold is usually not convergent, but in many cases the Taylor series for the bifurcating solutions do converge (assuming of course, that the nonlinear terms f and g have convergent Taylor expansions).
	www.math.vt.edu /people/renardym/class_home/nova/bifs/node18.html (582 words)

	MEJ Preprints
		The lower right panel shows a view of the stable manifold of the origin at c=7; notice the relative location of the upper side of the unstable manifold of the origin as well as its ``wrapping around" the stable manifold of the ``other" fixed point C. See text for further discussion.
		The lower left frame (c) shows the stable manifolds of each of a pair of coexisting saddle-type limit cycles in blue (and red) and part of one side of their unstable manifold in yellow (and olive green).
		Frame (a) shows one side of the stable manifold of each of two saddle-type limit cycles (in red) along with the one-dimensional unstable manifold of the origin (in green).
	www.ima.umn.edu /~mjohnson/preprints/anm.html (652 words)

	Manifold with boundary (Site not responding. Last check: 2007-10-14)
		An extension of manifold boundary representations to the r-sets...
		[hep-th/9603142] Eleven-Dimensional Supergravity on a Manifold with Boundary...
		Recovery of a Manifold with Boundary and its...
	www.scienceoxygen.com /math/665.html (226 words)

	The stable manifold theorem
		The stable manifold theorem says that the nonlinear system behaves in a qualitatively similar fashion; the only difference is that the linear subspaces must be replaced by curved manifolds.
		The existence of an unstable manifold is one avenue towards proving that linear instability implies nonlinear instability.
		There is also a version of the stable manifold theorem which covers the case where some of the eigenvalues are on the imaginary axis.
	www.math.vt.edu /people/renardym/class_home/nova/bifs/node16.html (454 words)

	The Stable Manifold Theorem For Stochastic Differential Equations - Mohammed, Scheutzow (ResearchIndex) (Site not responding. Last check: 2007-10-14)
		We formulate and prove a Local Stable Manifold Theorem for stochastic differential equations (sde's) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments.
		We prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary...
		0.7: The Stable Manifold Theorem For Nonlinear Stochastic..
	citeseer.lcs.mit.edu /mohammed98stable.html (529 words)

	Computing Geodesic Level Sets on Global (Un)stable Manifolds of Vector Fields (Site not responding. Last check: 2007-10-14)
		The global dynamics of such a system is organized by the stable and unstable manifolds of the saddle points, of the saddle periodic orbits, and, more generally, of all compact invariant manifolds of saddle type.
		Stable manifolds are computed by considering the flow for negative time.
		The properties and performance of our method are illustrated with several examples, including the stable manifold of the origin of the Lorenz system, a two-dimensional stable manifold in a four-dimensional phase space arising in a problem in optimal control, and a stable manifold of a periodic orbit that is a Möbius strip.
	epubs.siam.org /sam-bin/dbq/article/60018 (518 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		When large disturbances are considered, the existence of a stable fast dynamics equilibrium after a disturbance is not the only requirement for a valid time scale decomposition: The pre-disturbance state of the system must also belong to the region of attraction of the post disturbance stable equilibrium of the fast dynamics.
		When the %HB is subcritical, the boundary of the region of attraction of the stable points %near the bifurcation includes the stable manifold of the unstable limit cycle.
		The fast subsystem is stable on the upper part of the fast dynamics equilibrum manifold and unstable on the lower part of the fast dynamics equilibrum manifold.
	eceserv0.ece.wisc.edu /~dobson/May15WG/section7 (1614 words)

	5A1352 - Log 04
		The different stability types in 2-D: stable and unstable nodes and spiral nodes, saddles, and centers.
		A stable manifold cannot intersect itself or a stable manifold belonging to another FP, the same holds for unstable manifolds.
		A sketch of what happens when the unstable manifold of the origin returns to intersect the stable manifold of the origin.
	courses.physics.kth.se /5A1352/log04.html (1097 words)

	Patent 5792430: Solid phase organic synthesis device with pressure-regulated manifold (Site not responding. Last check: 2007-10-14)
		The manifold includes a pressure port coupled to both an inert gas source and a pressure control device, and a vacuum port coupled to a vacuum control device and a vacuum source.
		The manifold includes a pressure port coupled to both a regulated inert gas source and a means for controlling pressure within the manifold, and a vacuum port coupled to a means for controlling vacuum within the manifold.
		The top cover or second manifold 63 is sealed with a gasket 64 which sits partially in a groove extending upward into the walls of the box of manifold 63.
	www.freepatentsonline.com /5792430.html (5196 words)

	The Lorenz system (Site not responding. Last check: 2007-10-14)
		We show both manifolds in the pictures, but our main goal is to explore the geometry of the stable manifold and how it interacts with the attractor.
		The one-dimensional unstable manifold is computed with DsTool, using 50 initial points and 20,000 iterations.
		A series of pictures of the stable manifold as it is rotated about the z-axis.
	www.enm.bris.ac.uk /staff/hinke/vectorfields/lorenz/index.html (184 words)

	stable manifold theorem (Site not responding. Last check: 2007-10-14)
		That is the case when the unstable and the stable manifold of a (saddle-type) fixed point have transversal intersections.
		In this case it is conjectured, that the unstable manifold forms an attractor with "chaotic" dynamics.
		Hi all, I'm presenting the stable manifold theorem for non-linear ode's in an upcoming seminar (I'm a grad student).
	www.thehelparchive.com /new-2425538-277.html (650 words)

	Verification of Wada Basin Boundaries
		Since forward iterates of the boundary remain in the scattering region forever, the boundary set and the stable manifold of the invariant set are the same.
		This is due to time-reversal symmetry, that is, the unstable manifold can be obtained from the stable manifold by simply reversing the direction of the trajectories.
		7(b), the slope dz/ds of the stable manifold of the invariant set is negative, we have that L
	www-chaos.umd.edu /publications/wadabasin/node4.html (1121 words)

	Stability and Bifurcation
		In the time-continuous case, this stability area is the half-plane left of the imaginary axis, whereas in the time-discrete case it is the unit circle around the origin.
		Saddles and their stable manifolds are usually the boundaries of a basin of attraction.
		It helps to reduce the dimensionality of the phase space to the dimensionality of the so-called center manifold which in the bifurcation point is tangentially to the eigenspace of the marginal modes of the linear stability analysis (i.e., the eigenfunctions which neither decay nor expand exponentially).
	monet.physik.unibas.ch /~elmer/pendulum/bif.htm (1853 words)

	Applied Math T-shirt 1999
		The stable manifold of a fixed point is the set of points that are eventually mapped into that fixed point.
		As a system parameter is changed the number of "loops" in the manifold decreases, the parameter for the T-shirt pictures is close to the situation where the maximal number of loops exist.
		Distance along the invariant manifold is measured in terms of the number of iteration steps that are needed to get to a certain segment of the manifold close to the fixed point.
	amath.colorado.edu /department/Tshirt/1999 (555 words)

	Preprints (Site not responding. Last check: 2007-10-14)
		The stable manifold theorem for semi-linear stochastic evolution equations and stochastic partial differential equations, II: The existence of stable and unstable manifolds.
		These results give smooth stable and unstable manifolds in the neighbourhood of a hyperbolic stationary solution of the underlying stochastic equation.
		The stable and unstable manifolds are stationary, live in a stationary tubular neighbourhood of the stationary solution and are asymptotically invariant under the stochastic semiflow of the see/spde.
	www.lboro.ac.uk /departments/ma/preprints/papers03/03-22abs.html (169 words)

	Math in the Media from the AMS (Site not responding. Last check: 2007-10-14)
		The wire looping through the origin is the strong stable manifold of the system.
		The manifold's vertical axis of symmetry can be seen as a diagonal across the upper half of this image.
		The crocheted Lorenz manifold struck the fancy of the international media, including the BBC (Mathematicians crochet chaos), CBC Radio (Crocheting Chaos), the Austrian ORF, and Channel One in Russia.
	www.ams.org /mathmedia/archive/02-2005-media.html (729 words)

	[No title]
		Each point $x_s$, $y_s$ of the fast dynamics equilibrium manifold is the equilibrium point of a fast subsystem defined as: $$\epsilon\dot{y}_f=g(x_s,y_s+y_f)\eqno(\hbox{2.7-1})$$ where $y_f=y-y_s$ is the fast component of $y$.
		The fast subsystem is stable on the upper part of the fast dynamics equilibrium manifold and unstable on the lower part of the fast dynamics equilibrium manifold.
		If $\epsilon$ is assumed sufficiently small, the fast dynamics are approximated by vertical lines moving towards stable points of the fast dynamics equilibrium manifold and away from unstable points of the fast dynamics equilibrium manifold.
	eceserv0.ece.wisc.edu /~dobson/WG/section7 (1668 words)

Try your search on: Qwika (all wikis)