Mathematical Sciences: Numerics for Dynamics

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Mischaikow Most numerical methods for studying ordinary differential equations are based on one of the following two ideas: (1) tracking individual trajectories via an approximation of the differential equation, or (2) searching for orbits with predetermined properties, e.g. a periodic orbit or an invariant torus, by solving an associated boundary value problem. While these techniques are obviously successful, they have shortcomings; one can only approximate orbits over finite time intervals and these methods have difficulty detecting orbits which are unstable in the sense of dynamics. These unstable orbits can be extremely important since they define the boundaries of basins of attraction, and hence, determine the global dynamics of the system. In addition, there are many problems where the objects of interest are precisely the unstable bounded orbits, e.g. traveling waves. In the second case, one needs to know a priori what types of solutions should be looked for and in what regions of phase space one should look for the solutions. The investigator develops complimentary methods for studying the global dynamics of ordinary differential equations. The theoretical basis for the project is Conley's theory of dynamical systems. The work is separated into four tasks. (1) Given as an input an ordinary differential equation and a region in phase space, develop efficient numerical schemes to determine isolating neighborhoods and compute their Conley indices. (2) Given a potential isolating neighborhood, develop more efficient mechanisms for obtaining computer assisted proofs to verify that the neighborhood is an isolating neighborhood and to rigorously compute the Conley index of the associated isolated invariant set. (3) Extend the abstract Conley index theory, which allows one to draw conclusions concerning the structure of the dynamics from knowledge of the index and the isolating neighborhood. (4) Apply these techniques to ordinary differential equations for which the global dynamics is poorly understood. It should be emphasized that the end goal of these four steps is to obtain a rough understanding of the global dynamics of the ordinary differential equation. In particular, these procedures provide information about both the type and structure of dynamics that occur, and the region of phase space in which these dynamics take place. A finer understanding of these dynamics can then be more efficiently investigated using the more classical methods mentioned at the beginning. It is clear that most phenomena, whether physical, chemical or biological in nature, need to be described by nonlinear models. Thus, to describe the dynamics of these phenomena one is required to solve problems in global analysis. In practice, this usually requires numerical computations, since our theoretical analytic techniques are not yet adequate. With the advent of high performance computing considerable progress has been made in understanding the behavior of a wide variety of nonlinear dynamical systems. However, there are many drawbacks to most current numerical and computational techniques at our disposal. This project is aimed at two of them. The first arises from the fact that the classical numerical techniques often provide bad approximations over long time intervals (i.e. they are unreliable for long range predictions) and also they have difficulty detecting unstable dynamic structures (these unstable structures determine the path by which the object being modelled reaches its asymptotic state). The second problem involves difficulty in understanding the tremendous amounts of numerical data which the computers are capable of generating. Typically this problem is overcome by graphically displaying the data. Unfortunately, most models involve more than three variables, and hence visualizing the numerical results becomes a difficult task. This project develops theoretical techniques and efficient numerical methods for describing the global dynamics arising from nonlinear ordinary differential equations, and then applies these techniques to a wide range of biological and physical problems. The theoretical aspect involves translating algebraic information into dynamics and the numerical side involves computing these algebraic invariants by approximating the differential equations. The algebraic information works equally well for stable or unstable dynamics and, of course, being algebraic, does not need to be visualized in order to be understood.