Numerical Analysis of Qualitative Dynamics in Flows

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Kalies9973331The investigator develops numerical methods for determining the qualitative dynamical behavior of solutions to ordinary differential equations. For these methods, input requires only a system of differential equations, a region of the phase space, and no a priori knowledge of the dynamics. The output is a Morse decomposition of the invariant set into regions containing nontrivial dynamics. Information about the behavior in these regions is given in terms of the Conley index, and connecting orbits between Morse sets can be detected. Such an algorithm is beneficial as a first step in the study of systems of differential equations for which little information is known, particularly in higher dimensions where visualization of numerically simulated solutions is difficult. The type of computation envisioned here provides a somewhat coarse, qualitative description of a dynamical system. However, the Conley index, a tool from dynamical systems theory, is an algebraic topological index, which also allows for the development of algorithms for rigorously proving the existence of specific dynamical behavior. Therefore, a reliable algorithm can be used both as a tool for analyzing unfamiliar systems and as a tool for obtaining computer-assisted proofs of specific dynamics; both would have a wide variety of applications. The algorithms developed here also have potential applications in computational topology, in particular the computation of the homology of cellular complexes, and connections to mesh generation and Delaunay triangulations.Differential equations are used to model many phenomena in the physical sciences: mechanical systems, chemical reactions, and biological systems. Understanding the dynamics of such inherently nonlinear systems is a problem in global analysis. When the underlying space has dimension higher than three, the study of such systems is often beyond the scope of current analytic techniques. Computers are capable of performing many calculations of solutions to such equations, but most current numerical methods are not well suited for extracting asymptotic dynamical behavior from systems in higher dimensions. One problem is that computers can generate a large quantity of data that is difficult to analyze; often a graphical analysis is performed, which is impractical in higher dimensions and can miss unstable objects. The purpose of this project is to design and implement algorithms to study the asymptotic dynamics of differential equations in higher dimensions. The output of the algorithm is an algebraic index (designed by C. Conley) that could be thought of as a tool for extracting the essential dynamical information (including unstable objects) from computational data and that does not require visualization to be understood.