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...

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

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

manifold, stable manifold, unstable manifold, subcenter manifold and inertial manifold. Typically, although by no means always, invariant manifolds are constructed...

bifurcation, the stable manifold converges to the sink, and the unstable manifold escapes to infinity. After the event, the stable manifold converges to the...

The basin boundaries form a Cantor set and represent members of the stable manifold: trajectories that, once started, never exit the system. So long as...

arise from the linearizations of the stable manifold and unstable manifold of the fixed point. The unstable manifold of the fixed point in the attractor...

are adjacent Invariant set, said to be "stable" under a mapping or transformation Stable manifold or stable set, in dynamical systems, the set of points...

precisely, a homoclinic orbit lies in the intersection of the stable manifold and the unstable manifold of an equilibrium. It is a heteroclinic orbit–a path between...

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

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

about a neighborhood of a hyperbolic point, notably A stable manifold and an unstable manifold exist, Shadowing occurs, The dynamics on the invariant...

formed by the stable manifold and unstable manifold of a fixed point. Let f : M → M {\displaystyle f:M\to M} be a map defined on a manifold M {\displaystyle...

if x belongs to the interior of its stable manifold, it is asymptotically stable if it is both attractive and stable. (There are examples showing that attractivity...

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

Cycle detection Hyperbolic set Stable manifold Steady state Wada basin Hidden oscillation Rössler attractor Stable distribution Convergent evolution...

proves that NHIMs possess stable and unstable manifolds and more importantly, NHIMs and their stable and unstable manifolds persist under small perturbations...

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

Local sampling on image manifolds using diffusion models". arXiv:2210.12100 [cs.CV]. Bühlmann, Matthias (September 28, 2022). "Stable Diffusion Based Image...

short time-scales (hundreds of thousands of years), Saturn's rings are stable, and are thus a reasonable example of a conservative system and more precisely...

dimension of the stable manifold of a periodic point or fixed point is zero, the point is called a source; if the dimension of its unstable manifold is zero,...

fractional transformation decomposes complex projective space into stable and unstable manifolds, with the horocycles appearing perpendicular to the geodesics...

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

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...

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

satisfying a transversality condition on the stable and unstable manifolds. Morse–Smale systems are structurally stable and form one of the simplest and best...

1093/qmath/27.1.63. Brown, Morton; Gluck, Herman (1964), "Stable structures on manifolds. II. Stable manifolds.", Annals of Mathematics, Second Series, 79 (1):...

converge towards the orbit (the stable manifold) and another of the points that diverge from the orbit (the unstable manifold). This branch of mathematics...

implies that the orbit is contained in the stable manifold of x 1 {\displaystyle x_{1}} and the unstable manifold of x 0 {\displaystyle x_{0}} . By using...

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