In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, especially in the study of dynamical system

Dynamical system

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

s and differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s, the stable manifold theorem is an important result about the structure of the set of orbits

Orbit (dynamics)

In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. The orbit is a subset of the phase space and the set of all orbits is a partition of the phase space, that is different orbits do not intersect in the...

 approaching a given hyperbolic fixed point.

Stable manifold theorem

Let
be a smooth map with hyperbolic fixed point at p. We denote by the stable set

Stable set

The term stable set may refer to:* The independent set in graph theory, the set of vertices of a graph not directly connected by edges.* The stable set in dynamical systems, the set of points leading up to an attractor....

 and by the unstable set of p.

The theorem states that

• is a smooth manifold and its tangent space
  Tangent space
  In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
  
   has the same dimension as the stable space of the linearization
  Linearization
  In mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or...
  
   of f at p.
• is a smooth manifold and its tangent space has the same dimension as the unstable space of the linearization of f at p.

Accordingly is a stable manifold

Stable manifold

In mathematics, and in 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 notions embodied in the idea of an attractor or repellor...

and is an unstable manifold.