A new topological technique for characterizing homoclinic tangles

We have developed a topological approach, called homotopic lobe dynamics, for describing the qualitative structure of homoclinic tangles. This approach begins from an efficient and accurate description of the initial development of a tangle, up to some finite number of iterates J, where the value of J indicates the amount of information that one puts into the theory. Our approach can then compute the topologically forced structure of the tangle at all iterates after J. This allows one, for example, to predict a minimal set of homoclinic intersections. This technique places few assumptions on the homoclinic tangles considered. In fact, one main advantage is its ability to describe the wide variety of behaviour seen in physically significant tangles.