Lorenz System*

This example helps illustrate that this approach turns numerical instability into stability, by using a forward method in the stable direction normal to the manifold, and a backward method in the unstable direction normal to the manifold. The Lorenz system is a low order model of atmospheric circulation.  It is well-known for its chaotic attractor. At r=20.0, this system has a saddle-type periodic orbit.  For decreasing r, this periodic orbit tends toward a homoclinic orbit.  In this example, both the initial closed curve and the initial hyperbolic splitting were determined by simulation, hence were only roughly known.  We continued the periodic orbit toward the homoclinic orbit.  Also illustrated below is the computed hyperbolic splitting.

 

Periodic orbit of the Lorenz system, r=20.0, discretized with 200 first order elements.

 

Periodic orbit of the Lorenz system, r=16.5, 200 first order elements.

 

 

 

 

Hyperbolic Splitting of Periodic orbit of the Lorenz system, r=16.5,
discretized with 200 first order elements.

The user-written program which uses the class library to perform this computation is provided as a class library usage  example .

3D-fattened Arnold map**

For small positive epsilon, this map has a saddle-type invariant closed curve.  The dynamics on the curve are quasiperiodic with phase locking (thus the dynamics on the curve are determined by either a dense quasiperiodic orbit or a finite number of fixed points).  At about epsilon=0.49, there is a saddle-node bifurcation, and a sink and a source appear on the invariant curve.  Normal hyperbolicity is lost at about epsilon=0.776.  In spite of the changing dynamics on the invariant curve, the continuation illustrated below went very smoothly. Also visible in the pictures below is the computed hyperbolic splitting.

 

Closed curve of the fattened Arnold family, epsilon=0.0, discretized with 50 third order elements.

 

 

Closed curve of the fattened Arnold family, epsilon=0.776, discretized with 50 third order elements.

 

*C. Sparrow, ``The Lorenz equations: bifurcations, chaos, and strange attractors,'' Springer-Verlag, New York, 1982.

**H. M. Osinga, ``Computing Invariant Manifolds,'' Ph.D. Thesis, University of Groningen, The Netherlands, 1996.