Invariant manifold

In dynamical systems, a branch of mathematics, an invariant manifold is a topological manifold that is invariant under the action of the dynamical system. Examples include the slow manifold, center manifold, stable manifold, unstable manifold, subcenter manifold and inertial manifold.

Typically, although by no means always, invariant manifolds are constructed as a 'perturbation' of an invariant subspace about an equilibrium. In dissipative systems, an invariant manifold based upon the gravest, longest lasting modes forms an effective low-dimensional, reduced, model of the dynamics.

Consider the differential equation ${\displaystyle dx/dt=f(x),\ x\in \mathbb {R} ^{n},}$ with flow ${\displaystyle x(t)=\phi _{t}(x_{0})}$ being the solution of the differential equation with ${\displaystyle x(0)=x_{0}}$. A set ${\displaystyle S\subset \mathbb {R} ^{n}}$ is called an invariant set for the differential equation if, for each ${\displaystyle x_{0}\in S}$, the solution ${\displaystyle t\mapsto \phi _{t}(x_{0})}$, defined on its maximal interval of existence, has its image in ${\displaystyle S}$. Alternatively, the orbit passing through each ${\displaystyle x_{0}\in S}$ lies in ${\displaystyle S}$. In addition, ${\displaystyle S}$ is called an invariant manifold if ${\displaystyle S}$ is a manifold.

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