# Global stable manifold of the Lorenz system

### Invariant Manifolds

By Bernd Krauskopf & Hinke Osinga
Print

## Invariant manifolds of the Lorenz system

The Lorenz system is the three-dimensional vector field

\begin{array}{rcl} \{\dot x &=& \sigma (y-x) \ \dot y &=& \varrho x - y -xz \ \dot z &=& xy - \beta z, \end{array}

We choose the standard parameter values parameters: $$\sigma=10$$, $$\rho=28$$, but take $$\beta=0.4$$. The two-dimensional stable manifold of the origin interacts now with an attracting periodic orbit. The colored bands on the manifold show the steps that are taken by the algorithm. Note how the algorithm slows down each time a new helix around the $$z$$-axis is being formed.

 Author Institutional Affiliation University of Bristol Author Email [email protected] Notes This entry was not submitted by the original authors, but by the Picture gallery editors. Keywords Smooth Dynamical Systems, SIADS
Tags: