Mathematical Sciences: Numerical Solution of Matrix Differential Equations and Approximation of Invariant Tori

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9306412 Dieci Luca Dieci will work in the area of computational equations. The PI and his collaborators will undertake several projects, centered around the following two: (1). Numerical Solution of Matrix Differential Equations; (2). Numerical Approximation of Invariant Tori. More in particular, Luca Dieci expects to continue working on differential Riccati equations (DREs) and on applications related to these equations. Algorithmical and theoretical work is specifically planned for the numerical solution of symmetric DREs, a common occurrence in Engineering applications. The main effort here will be in studying stable indirect solution strategies for the DREs, rather than direct discretizations. Luca Dieci intends also to continue studying numerical questions connected to the integration of matrix equations whose solution is a unitary matrix. He plans to devote time and resources to the approximation of Lyapunov exponents, an issue which provided the motivation for studying matrix equations with a unitary solution. Finally, he intends to continue the work on approximation of invariant tori dynamical systems. In particular, he expects to extend and improve upon the previously studied PdE approach. Here, the main directions of research will involve theoretical and practical study in order to complete the approximation theory for the nonlinear case, and in order to free the numerical procedures from the explicit knowledge of a coordinate system in which to parametrize the tori. The main plan consists in using the continuous theory of Fenichel and in making it into a numerical procedure, and in carefully studying the associated implementation details. The above projects have a strong computational component and aim at developing reliable and rigorously justified algorithms for important problems in computational ODEs. ***