Mathematical Sciences: Some Approximation Problems in Differential Equations

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9625813 Dieci The investigator and his collaborators analyze and implement techniques to resolve outstanding computational issues in differential equations. The following topics are studied: (1) computation and bifurcations of invariant tori, (2) computation of Lyapunov exponents and applications, (3) orthogonal factorizations for time-dependent matrices and applications, (4) Riccati equations, (5) logarithms of matrices and matrix interpolation. In each case the project combines theoretical analysis and extensive computational testing, and develops working codes. Several aspects of this research are being carried out in collaboration with Timo Eirola, Jens Lorenz, Erik Van Vleck, Alessandra Papini, Aldo Pasquali, and Ph.D. students Michael Keeve and Benedetta Morini. The investigator analyzes and implements algorithms to compute invariant tori of parameter-dependent dynamical systems and to study their bifurcations. He studies ways to monitor Lyapunov type numbers on the tori in order to detect their smoothness and understand and classify their bifurcations and breakdown mechanisms. The whole issue of computation of Lyapunov type numbers (or Lyapunov exponents) is also considered. Quantitative information on these numbers is needed in order to obtain a reliable stability picture for time-dependent linear systems and nonlinear systems. He examines applications of this tool to error control. Efficient and reliable techniques for computation of Lyapunov exponents require study of orthogonal factorizations of time-dependent matrices. The investigator works on numerical solution of Riccati equations, and develops an integration code based on Gauss Runge-Kutta schemes. He studies techniques for computation of logarithms of matrices, emphasisizing the case of a sequence of slowly varying matrices and its application to interpolants of matrices. Finally, he is writing a book on numerical dynamical systems with emphasis on direct approximation techniques . Differential equations are one of the most powerful models to describe physical phenomena. They provide a compact description of a system by giving a continuous description of how a given position in (phase) space of a system at a particular time influences neighboring positions in the immediate future. The price for such a compact description is that we then need to solve the differential equation in order to obtain information about the system. Solving the differential equation, except in trivial cases, cannot be done exactly, and we must resort to numerical techniques. The investigator devises numerical techniques for differential equations, motivated by the following question: What should we approximate, what should we monitor? Existing techniques are typically very reliable at tracking a specific solution (a trajectory) of the equation over a short time interval, but they are inadequate to provide a complete description of the system, which would need knowledge about many solutions over long time intervals. This need is particularly true for models of complicated phenomena, which may take the form of systems of coupled oscillators and coupled nonlinear differential equations, or high-dimensional time-varying matrix equations. These cases are of interest in this project and find applications in almost all applied sciences: from the study of complicated biological and chemical interactions, to the optimal control of time-dependent manufacturing processes, to stability assessment for time-dependent phenomena in general. In concrete, the following issues are studied. (1) For many systems, all solutions eventually approach a finite region of phase space, and then stay there afterwards. The investigator devises techniques that target directly such eventual finite regions, to avoid transient behavior. (2) A differential equation, because it mimics a given physical system, usually has built in some key properties. For example, in modeling a rigid rotat ion, the solutions of the differential equation should be rigid rotations. Unfortunately, most numerical techniques will fail at preserving even such a seemingly simple characteristic. The failure of a numerical method to preserve essential characteristics might have very unpleasant effects. The investigator works on techniques that maintain the relevant geometrical properties of the solutions. (3) For a given model, the fundamental question is the one of stability. How robust is the model? How sensitive are solutions to small changes in the model? Or, in a deceivingly different but mathematically equivalent way, how fast is a system approaching its eventual state? The investigator studies ways to monitor indicators in order to answer these questions, and also applies this study to assess error propagation during discretization.