TOPICS

# Invariant Manifold

An invariant set is said to be a () invariant manifold if has the structure of a differentiable manifold (Wiggins 1990, p. 14).

When stable and unstable invariant manifolds intersect, they do so in a hyperbolic fixed point (saddle point). The invariant manifolds are then called separatrices. A hyperbolic fixed point is characterized by two ingoing stable manifolds and two outgoing unstable manifolds. In integrable systems, incoming and outgoing manifolds join up smoothly.

Homoclinic Point

## Explore with Wolfram|Alpha

More things to try:

## References

Rasband, S. N. "Invariant Manifolds." §5.2 in Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 89-92, 1990.Wiggins, S. "Invariant Manifolds: Linear and Nonlinear Systems." §1.1C in Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer-Verlag, pp. 14-25, 1990.

## Referenced on Wolfram|Alpha

Invariant Manifold

## Cite this as:

Weisstein, Eric W. "Invariant Manifold." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InvariantManifold.html