Our current research effort is reaching a conclusion because our goals have been limited and focused.
Our present goals are to create a kind of analog for Fourier analysis, but applicable to nonlinear dynamical systems. This goal is nearing completion because we have focused on dissipative dynamical systems in low dimensional spaces (three dimensions, to be precise).
We have found that there are topological structures that can be used to describe and classify strange attractors. These can be identified by a set of integers (we are doing topology, after all). The information required to carry out this classification can be extracted from experimental data, and we have developed algorithms for determining this information from experimental data.
It turns out that four levels of (topological) structure are required to classify strange attractors in low dimension. These are:
To make a long story short, strange attractors in three-space can be built up, Lego-like, from two building units: one describing stretching and the other describing squeezing. The stretching units are responsible to sensitivity to initial conditions and the squeezing units are responsible for keeping the attractor bounded in phase space. In this sense building up strange attractors is like being children playing with Lego building blocks.
We need to extend this topological description of strange attractors into higher dimensions (than three). The sticky point here is to construct topological invariants in four- and higher dimensions that can be extracted from the unstable periodic orbits that exist in large numbers in strange attractors. In three dimensions we used the Gauss linking number. We are assured by many mathematicians that extending the Gauss linking number into higher dimensions is a hopeless task --- the more we are told that the harder we try. If/once this is accomplished, the Gauss index (whatever that turns out to be) will be used to distinguish among the different types of foldings that can occur to creat strange attractors in higher dimensions.
This is our current goal: to develop new tools and to test these new tools out on experimental data generated by higher dimensional strange attractors.
--R. Gilmore, January 2005