Unstable manifold - Factbites

Unstable manifold - Factbites
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Topic: Unstable manifold

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	Arneodo's system
		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.
		This means that one of its Floquet multipliers moved through -1 and the unstable manifold is, therefore, non-orientable.
	www.maths.ex.ac.uk /~hinke/nonorientable (259 words)

	Linear 2D systems
		Except for degenerate cases, the sum of the dimensions of the unstable and stable manifolds is equal to 2, the dimension of the plane.
		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.
		The unstable manifold for a saddle point is the eigenvector corresponding to the positive eigenvalue.
	www.cnbc.cmu.edu /~bard/xppfast/lin2d.html (259 words)

	99-273.latex
		Moreover the stable and unstable manifolds are of (equal) finite dimension and the center manifold is infinite-dimensional.
		The consequence is that stable, unstable, and center manifolds exist in the neighborhood of a (stable or unstable) standing wave, such as a waveguide mode, under simple and commonly verifiable spectral conditions.
		The (local) stable manifold is defined as the set of initial data whose solutions stay in the prescribed neighborhood and tend to $\hat{u}$ exponentially as $t \rightarrow +\infty$.
	www.ma.utexas.edu /mp_arc/e/99-273.latex (259 words)

	PlanetMath: hyperbolic fixed point
		If the dimension of the stable manifold of a fixed point is zero, the point is called a source ; if the dimension of its unstable manifold is zero, it is called a sink ; and if both the stable and unstable manifold have nonzero dimension, it is called a saddle.
		Cross-references: stable, stable manifold, dimension, iterate, least period, periodic point, linear hyperbolic isomorphism, diffeomorphism, fixed point, smooth manifold
		This is version 3 of hyperbolic fixed point, born on 2003-07-27, modified 2003-07-29.
	planetmath.org /encyclopedia/HyperbolicFixedPoint.html (259 words)

	Research Papers from the Geometry Center
		The algorithm computes intersections of the unstable manifold with a finite number of leaves of a chosen linear foliation.
		This electronic document presents an algorithm for computing the global two-dimensional unstable manifold of a normally hyperbolic invariant circle of a three-dimensional map.
		This technique is used to complete the proof of several long-standing rigidity conjectures in 3-manifold theory as well as to provide a new lower bound for the volume of a closed orientable hyperbolic 3-manifold.
	www.geom.uiuc.edu /docs/research (259 words)

	Defect nucleations - Defect mediated turbulence
		There are indications that the unstable manifold of the attractor in spiral defect chaos is centered on events associated with defect nucleation.
		The trajectory of a system at a chaotic attractor can be viewed as being scattered by the unstable manifolds of fixed points and periodic orbits while the stable manifold keeps the system bounded.
		We have developed a procedure to extract the linear manifold about fixed points and periodic orbits for spatially extended dynamical systems.
	cns.physics.gatech.edu /~kapil/research/defects.html (259 words)

	501-general.html
		Any horizontally homothetic harmonic morphism to an unstable manifold is unstable, [Montaldo 1996]
		Any harmonic morphisms from a manifold with boundary to a complete Riemann surface with a Hermitian metric h, whose Gauss curvature is bounded from above, is stable with respet to some metric $h'\in[h]$, [Montaldo 1996]
		Any submersive harmonic morphism with totally geodesic fibres to a surface from a manifold with non positive curvature is stable, [Montaldo 1996]
	riemann.unica.it /~montaldo/homepage/atlas/501-general.html (259 words)

	Citations: Bifurcation of degenerate homoclinics - Vanderbauwhede (ResearchIndex)
		In particular we assume that the homoclinic orbit fl is degenerate that is: along fl the stable and unstable manifolds of the equilibrium have an intersection which is (at least) two dimensional.
		....We work in the rst chart again and use a distance function d(1 ; s 1) measuring the distance of the unstable manifold of p r and the stable manifold of p a at u 1 = 0.
		Starting from the same suppositions on (3) as Vanderbauwhede did, we show that the (ffl) set for which the periodically forced system (2 ;ffl) has homoclinic points is foliated by....
	citeseer.ist.psu.edu /context/266589/0 (1097 words)

	Center Manifold Theorem
		Numerical methods for the construction of the unstable and stable manifolds are described in reference [8].
		The invariant manifolds of a flow are composed from a union of solution curves; the invariant manifolds of a map consist of a union of a discrete collection of points (Fig.
		These invariant manifolds are a direct generalization of the invariant subspaces of the linear problem.
	www.drchaos.net /drchaos/Book/node119.html (447 words)

	PlanetMath: stable manifold
		Cross-references: hyperbolic set, 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.
		This result is also valid for nonperiodic points, as long as they lie in some hyperbolic set (stable manifold theorem for hyperbolic sets).
	planetmath.org /encyclopedia/StableManifold.html (447 words)

	Arneodo: animation
		The unstable manifold of the periodic orbit for
	www.maths.ex.ac.uk /~hinke/nonorientable/perwu.html (17 words)

	Matrix manifold
		IngentaConnect The Evolution of the Stable and Unstable Manifold of an Equilibri...
		Manifold Pursuit: A New Approach to Appearance Based Recognition...
		Numerical solution of ODEs on the oblique rotation matrix manifold via the corre...
	www.scienceoxygen.com /math/702.html (270 words)

	98-412
		In the case of an $ n $-dimensional Anosov manifold $ M $, the stable and unstable distributions are defined on the manifold $ \Omega M $ of unit tangent vectors.
		The class of Anosov manifolds is opposite in some sense to the class of homogeneous Riemannian manifolds because the isometry group of an Anosov manifold is discrete \cite{Es}.
		For such a manifold, the set of closed geodesics is dense in the set of all geodesics.
	www.ma.utexas.edu /mp_arc/html/papers/98-412 (4479 words)

	Publications
		Derks and T. Ratiu [2002] Unstable manifolds of relative equilibria in Hamiltonian systems with dissipation Nonlinearity 15, 531-549.
		[2001] Computing Lyapunov exponents on a Stiefel manifold.
	www.maths.surrey.ac.uk /research/General/publications.html (4479 words)

	SMOOTH INVARIANT MANIFOLDS AND NORMAL FORMS
		Its main purpose is to present, as completely as possible, the basic results concerning the existence of stable and unstable local manifolds and the recent advancements in the theory of finitely smooth normal forms of vector fields and diffeomorphisms in the vicinity of a rest point and a periodic trajectory.
		This book deals with the qualitative theory of dynamical systems and is devoted to the study of flows and cascades in the vicinity of a smooth invariant manifold.
		A summary of the results obtained so far in the investigation of dynamical systems near an arbitrary invariant submanifold is also given.
	www.worldscibooks.com /chaos/2184.htm (154 words)

	Collapse Theories
		The statevector is supposed to undergo random processes at random times, inducing sudden changes driving it either within the linear manifold of the unstable state or within the one of the decay products;
		Grassi and myself [Ghirardi and Grassi, 1991] showed that stochastic modifications without nonlinearity can at most induce ensemble and not individual reductions, i.e., they do not guarantee that the state vector of each individual physical system is driven in a manifold corresponding to definite properties.
		Gisin gave subsequently an extremely interesting proof [Gisin, 1989] that nonlinear modifications of the standard equation without stochasticity are unacceptable since they imply the possibility of sending superluminal signals.
	plato.stanford.edu /entries/qm-collapse (154 words)

	A Model for the Unstable Manifold of the Bursting Behavior in the 2D Navier--Stokes Flow
		Three eigenfunctions that represent the dynamics of the quasi-periodic regime and two eigenfunctions associated with the unstable manifold of the bursting regime were derived.
		Processing these time series through an ANN results in a low-dimensional model describing the unstable manifold of the bursting regime that can be used to predict the onset of a burst.
		Inverse Fourier transform is applied to represent the POD eigenfunctions in both streamfunction and vorticity formulations so that the number of relevant eigenfunctions for streamfunction and vorticity data is the same.
	epubs.siam.org /sam-bin/dbq/article/35501 (154 words)

	Collapse Theories
		To make the treatment quite general (the apparatus does not know which kind of unstable system it is testing) one is led to identify the random processes with localization processes of the relative coordinates of the decay fragments.
		> and the corresponding eigenmanifolds (the linear manifolds spanned by the eigenvectors associated to a given eigenvalue, also called eigenspaces) play a basic role for the predictive content of the theory.
		The fact that when the measurement is completed one can make statements about the outcome is accounted for by the already mentioned WPR postulate [Dirac, 1948]: a measurement always causes a system to jump in an eigenstate of the observed quantity.
	plato.stanford.edu /entries/qm-collapse (10427 words)

	Publications
		Derks and T. Ratiu [2002] Unstable manifolds of relative equilibria in Hamiltonian systems with dissipation Nonlinearity 15, 531-549.
		[2001] Computing Lyapunov exponents on a Stiefel manifold.
		R. Cushman and R.M. Roberts [2002] Poisson structures transverse to coadjoint orbits Bull.
	www.maths.surrey.ac.uk /research/General/publications.html (10427 words)

	resume
		n°145 Conjugate unstable manifolds of saturated hyperbolic sets of Smale diffeomorphisms are characterised in terms of the combinatorics of their geometrised Markov partitions.
		We also care about the special subriemannian metrics that are related to isoperimetric problems on a 2-d riemannian manifold.
		On the contrary, for the Dido problem on a 2-d Riemannian manifold (i.e.
	math.u-bourgogne.fr /monge/topol/prepub/resume.html (10427 words)

	Victoria - April 2001 Forecast Outlook, Discussion & Report Page
		The system had been made more than normally unstable with the effects of an upper level cold pool which had been fed in from the southwest in the previous few days.
		Basically the reason for the high totals in the region was the growth of a low pressure trough over eastern Victoria which developed a slow-moving wrap-around cloud band moving from the east and southeast over the Bellarine Peninsula, Geelong and the eastern Otways.
		I live in Heidelberg, which was on the northern boundary of the (near-stationary) rain area, and had the odd experience of spending most of a 100-minute run in sunshine and rain simultaneously (normally such conditions only last for a few minutes at most, in short-livedshowers).
	www.stormchasers.au.com /apr01.htm (10427 words)

	Software for maps: normally hyperbolic invariant manifolds
		local stable and unstable manifolds of invariant circles and tori.
		The BOV -method is an algorithm for the computation of normally hyperbolic invariant manifolds.
		The BOV -method is not influenced by changes of the dynamics on the manifolds, as long as they are normally hyperbolic.
	www.enm.bris.ac.uk /staff/hinke/dss/map/bov.html (10427 words)

	Publications
		Derks and T. Ratiu [2002] Unstable manifolds of relative equilibria in Hamiltonian systems with dissipation Nonlinearity 15, 531-549.
		[2001] Computing Lyapunov exponents on a Stiefel manifold.
		Melbourne, V. Nitica and A. T�r�k [2004] Stable transitivity of certain noncompact extensions of hyperbolic systems.
	www.maths.surrey.ac.uk /research/General/publications.html (10427 words)

	Report.html
		In particular we prove the universality of the graded derivation-based first-order differential calculus and show, that M (njm) is a \noncommutative graded manifold" in a stricter sense: There is a natural body map and the cohomologies of M (njm) and its body coincide (as in the case of ordinary graded manifolds).
		Abstract:The paper is devoted to the study of the relationship between integral manifolds of ordinary diöerential equations and duck>=trajectories.
		We derive suÖcient conditions for the existence of continuous slow integral surfaces that are devided into stable and unstable parts and propose a method of construction of surfaces consisting of duck>=trajectories.
	www.mathematik.uni-osnabrueck.de /projects/carmen/AP11/Report.html (10427 words)

	Differential and Riemannian Manifolds (Graduate Texts in Mathematics) Books
		In differential equations one studies vector fields and their integral curves, singular points, stable and unstable manifolds, and the like.
		This is the third version of a book on Differential Manifolds; in this latest expansion three chapters have been added on Riemannian and pseudo-Riemannian geometry, and the section on sprays and Stokes' theorem have been rewritten.This text provides an introduction to basic concepts in differential topology, differential geometry and differential equations.
		In differential geometry one adds structures to the manifold (vector fields, sprays, a metric, and so forth) and studies their properties.
	www.supermantv.net /0387943382/Differential_and_Riemannian_Manifolds_Graduate_Texts_in_Mathematics.html (352 words)

	chicone-tex
		To prove this, use the $\mathcal{C}^2$ stable manifold theorem to flatten the stable and the unstable manifolds onto the corresponding coordinate axes.
		A theorem from invariant manifold theory is used to ``flatten'' the invariant manifold corresponding to the block whose eigenvalues have the largest real part onto the corresponding linear subspace.
		It uses a version of the stable manifold theorem, consideration of the gaps in the spectrum of the linearized vector field at the rest point, carefully constructed Gronwall type estimates, and an induction argument.
	ejde.math.txstate.edu /Monographs/02/chicone-tex (6509 words)

	Differential and Riemannian Manifolds (Graduate Texts in Mathematics) Books
		In differential equations one studies vector fields and their integral curves, singular points, stable and unstable manifolds, and the like.
		This is the third version of a book on Differential Manifolds; in this latest expansion three chapters have been added on Riemannian and pseudo-Riemannian geometry, and the section on sprays and Stokes' theorem have been rewritten.This text provides an introduction to basic concepts in differential topology, differential geometry and differential equations.
		In differential geometry one adds structures to the manifold (vector fields, sprays, a metric, and so forth) and studies their properties.
	www.supermantv.net /0387943382/Differential_and_Riemannian_Manifolds_Graduate_Texts_in_Mathematics.html (176 words)

	s41d-22 in fm96
		We may then deduce that the global unstable manifold of the numerically obtained stationary solution, obtained by evolving the perturbed local unstable manifold forward in time, is lower semi-continuous, meaning that the true global unstable manifold is a subset of the perturbed unstable manifold in the limit as the numerical discretization size tends to zero.
		In case the global attractor is the closure of the unstable manifolds (of overflowing manifolds) such as with certain reaction-diffusion equations, we find that the attractor is lower semi-continuous as well.
		This research has continued in three main areas: large scale MHD simulations to deteremine the geometry and size of the upward going shock, theory to estabilish thresholds and efficiency of the shock, and particle simulations to examine the acceleration process in realistic magnetic and electric field structures found in collisionless shocks.
	www.agu.org /cgi-bin/SFgate/SFgate?&listenv=table&multiple=1&range=1&directget=1&application=fm96&database=/data/epubs/wais/indexes/fm96/fm96&maxhits=200&="S41D-22" (176 words)

	PlanetMath: hyperbolic fixed point
		If the dimension of the stable manifold of a fixed point is zero, the point is called a source; if the dimension of its unstable manifold is zero, it is called a sink; and if both the stable and unstable manifold have nonzero dimension, it is called a saddle.
		Cross-references: stable, stable manifold, dimension, iterate, least period, periodic point, linear hyperbolic isomorphism, diffeomorphism, fixed point, smooth manifold
		This is version 3 of hyperbolic fixed point, born on 2003-07-27, modified 2003-07-29.
	planetmath.org /encyclopedia/HyperbolicFixedPoint.html (176 words)

	Math 4791/5791 Solutions 3
		In the (u,v) coordinate system, trajectories approach the origin along the line u=0 (the stable manifold) and recede from the origin along the line v=0 (the unstable manifold).
		which means that the unstable manifold (just a fancy way of saying the direction along which trajectories escape) is y=-x.
		which means that the stable manifold (the direction along which trajectories approach the origin) is y=x.
	www-math.cudenver.edu /~wbriggs/4791s00/sol3/sol3/sol3.html (176 words)

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