Computation of Invariant Manifolds

Invariant manifolds are submanifolds of the phase space of a dynamical system that are unions of trajetories. Equivalently, a submanifold is invariant if the vector field of the system is tangent to manifold. Three types of invariant manifolds arising naturally in dynamical systems theory that have been studied computationally are invariant tori, stable/unstable manifolds of equilibria and periodic orbits and the slow manifolds of systems with multiple time scales. Alexander Vladimirsky and I developed a new class of algorithms for the computation of stable/unstable manifolds of equilibria. The images below come from this work. The one on the left is a depiction of  the two dimensional stable manifold of the origin in the Lorenz system. The paper XXX discusses several alternative methods for computing invariant manifolds, using this example to illustrate the output from each.  The image on the right depcits the two dimensional unstable of an equilibrium point in a four dimensional dynamical system modeling two pendula that are coupled by a torsional spring.
Stable manifold of origin in Lorenz system
The computation of invariant manifolds is an easy task conceptually: the manifold is swept out by trajectories of a suitable set of initial conditions. However, implementation of this process encounters several difficulties. Nearby trajectories may separate from one another sufficiently fast that the chosen trajectories do not enter large areas of the desired manifold. Adaptive methods for adding new initial conditions and new trajectories can ameliorate this problem if one formulates strategies for interpolation of new initial points. A second difficulty with algorithms based upon computing trajectories is that the speed of trajectories in different parts of the manifold may differ enormously. Unless the vector field is rescaled to prevent this, the result is that strips of the manifold along the fastest trajectories are computed long before the parts of the manifold along slow trajectories are obtained.  The algorithms that Vladimirsky and I created are based upon techniques adapted from methods used to solve first order partial differential equations. Instead of evolving the manifold along trajectories, we evolve the manifold as a triangulated object. New triangles are added successively to parts of the boundary where the vector field points outward, with the new vertex of the triangle  located so that the vector field will be tangent to the triangle at this vertex. This strategy resembles both " ordered upwinding methods" for solving partial differential equations and implicit methods for numerically integrating ordinary differential equations. Our method has both advantages and disadvantages: it is very fast but its order of accuracy is low.