the study of dynamical systems and differential equations, the stable manifold theorem is an important result about the structure of the set of orbits...

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...

neighbourhood. Linear approximation Stable manifold theorem Arrowsmith, D. K.; Place, C. M. (1992). "The Linearization Theorem". Dynamical Systems: Differential...

direction defines the stable manifold, the stretching direction defining the unstable manifold, and the neutral direction is the center manifold. While geometrically...

Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical...

topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, orientable, closed 4-manifold M has a spin structure (or, equivalently, the...

manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n {\displaystyle n} -dimensional manifold,...

equilibrium point (x*, y*) is called the community matrix. By the stable manifold theorem, if one or both eigenvalues of A {\displaystyle \mathbf {A} } have...

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a...

Alternately, conservative systems are those to which the Poincaré recurrence theorem applies. An important special case of conservative systems are the measure-preserving...

separability of the classes, and measures of geometry, topology, and density of manifolds. For non-binary classification problems, instance hardness is a bottom-up...

non-embedded) manifold with a given stable trivialisation of the tangent bundle. A related notion is the concept of a π-manifold. A smooth manifold M {\displaystyle...

Kähler manifold. The theorem can be considered a vast generalisation of the Narasimhan–Seshadri theorem which defines a correspondence between stable vector...

heat flow, using a convergence theorem for the flow to show that any smooth map from a closed manifold to a closed manifold of nonpositive curvature can...

opposite in sign. Hence by the stable manifold theorem, the equilibrium is a saddle point and there exists a unique stable arm, or “saddle path”, that converges...

correspondence (or Donaldson–Uhlenbeck–Yau theorem) relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. The correspondence...

mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold (e.g. differentiable...

a branch of mathematics, the stable normal bundle of a differentiable manifold is an invariant which encodes the stable normal (dually, tangential) data...

the annulus theorem in dimensions n ≥ 5 {\displaystyle n\geq 5} . It was also employed in further investigations of topological manifolds with Laurent...

The most common is the stable manifold or its kin, the unstable manifold. Ushiki's theorem was published in 1980. The theorem appeared in print again...

} The stable equivalence is an equivalence relation on semialgebraic subsets generated by stable projections and rational equivalence. Theorem (Mnëv's...

version of the transversality theorem. Let f : X → Y {\displaystyle f\colon X\rightarrow Y} be a smooth map between smooth manifolds, and let Z {\displaystyle...

formula for the area of a stable minimal hypersurface of a three-dimensional Riemannian manifold. The Gauss–Bonnet theorem then highly constrains the...

eigenvalues of the community matrix have negative real part, then by the stable manifold theorem the system converges to a limit point. Since the determinant is...

1093/imanum/drw020, MR 3649420. Epstein, C. L.; Weinstein, M. I. (1987), "A stable manifold theorem for the curve shortening equation", Communications on Pure and...

Riemannian manifolds, Darboux's theorem states that all symplectic manifolds are locally isomorphic. The only invariants of a symplectic manifold are global...

equations, vector fields on smooth manifolds and flows generated by them, and diffeomorphisms. Structurally stable systems were introduced by Aleksandr...

studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness...

equations. Attractor Hyperbolic set Periodic point Self-oscillation Stable manifold Thomas, Jeffrey P.; Dowell, Earl H.; Hall, Kenneth C. (2002), "Nonlinear...

intimately related to the transversality properties of smooth manifolds—see Thom transversality theorem. By reversing this construction, John Milnor and Sergei...