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...

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...

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

be divided into three parts: the stable manifold, the unstable manifold, and the center manifold. The stable manifold consists of all of the tangent vector...

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...

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...

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...

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

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...

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

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...

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

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

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

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,...

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

attractor comes into contact with an unstable periodic orbit or its stable manifold. As the orbit approaches the unstable orbit it will diverge away from...

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

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

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

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

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

self-dual. Every weakly stable Yang–Mills field over a compact orientable homogenous Riemannian 4 {\displaystyle 4} -manifold with gauge group SU ⁡ (...

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...

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

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...