Invariant manifold

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In dynamical systems, a branch of mathematics, an invariant manifold is a topological manifold that is invariant under the action of the dynamical system.[1] 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. [2]

Definition

Consider the differential equation dx/dt=f(x),\ x\in {\mathbb R}^{n}, with flow x(t)=\phi _{t}(x_{0}) being the solution of the differential equation with x(0)=x_{0}. A set S\subset {\mathbb R}^{n} is called an invariant set for the differential equation if, for each x_{0}\in S, the solution t\mapsto \phi _{t}(x_{0}), defined on its maximal interval of existence, has its image in S. Alternatively, the orbit passing through each x_{0}\in S lies in S. In addition, S is called an invariant manifold if S is a manifold. [3]

Examples

Simple 2D dynamical system

For any fixed parameter a, consider the variables x(t),y(t) governed by the pair of coupled differential equations

dx/dt=ax-xy\quad {\text{and}}\quad dy/dt=-y+x^{2}-2y^{2}.

The origin is an equilibrium. This system has two invariant manifolds of interest through the origin.

  • The vertical line x=0 is invariant as when x=0 the x-equation becomes dx/dt=0 which ensures x remains zero. This invariant manifold, x=0, is a stable manifold of the origin (when a\geq 0) as all initial conditions x(0)=0,\ y(0)>-1/2 lead to solutions asymptotically approaching the origin.
  • The parabola y=x^{2}/(1+2a) is invariant for all parameter a. One can see this invariance by considering the time derivative d/dt[y-x^{2}/(1+2a)] and finding it is zero on y=x^{2}/(1+2a) as required for an invariant manifold. For a>0 this parabola is the unstable manifold of the origin. For a=0 this parabola is a center manifold, more precisely a slow manifold, of the origin.
  • For a<0 there is only an invariant stable manifold about the origin, the stable manifold including all (x,y),\ y>-1/2.

See also

References

  1. ↑ Hirsh M.W., Pugh C.C., Shub M., Invariant Manifolds, Lect. Notes. Math., 583, Springer, Berlin — Heidelberg, 1977
  2. ↑ A. J. Roberts. The utility of an invariant manifold description of the evolution of a dynamical system. SIAM J. Math. Anal., 20:1447–1458, 1989. http://locus.siam.org/SIMA/volume-20/art_0520094.html
  3. ↑ C. Chicone. Ordinary Differential Equations with Applications, volume 34 of Texts in Applied Mathematics. Springer, 2006, p.34
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