| The work presented in this thesis continues to emphasize a new direction in the numerical analysis of dynamical systems. The shift in focus is accomplished by considering groups of trajectories. Briefly, using only information about the vectorfield, a simplicial or polygonal complex is constructed that is “aligned” with the flow. The underlying flow induces a multivalued map or graph on the complex. The complex and map serve to discretize the flow and essentially convert dynamics to combinatorics. The goal of this thesis is to understand what conditions on the cells guarantee that the induced map contains computable information about the dynamics.; In this thesis we present results that extend the previous planar ones to arbitrary dimensional Euclidean spaces. Also, by passing from simplicial complexes to polygonal ones we circumvent the thorny issue of transversality, and isolate its consideration to a portion of the boundary of the complex. Along these lines, we prove the result that any strongly connected component of our polygonal complex is an isolating block. This result ensures that we can extract Conley Index information from the induced map.; The central result of the thesis is a statement that if each simplex in the complex is δ-oriented with respect to the vectorfield then the complex ε approximates the chain recurrent set. This result says that if you can lay down an oriented complex, then you can extract as much, or better said as fine, dynamical information as is possible. Practically, this means that you can get as detailed an approximation, to a complete Lyapunov function on the maximal invariant set under consideration, as you are willing to pay for. |
