Computation of center manifold and normal form and application

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An important problem encountered in studying nonlinear dynamical systems is to find the governing equation of a system, which can be used to analyze important dynamical behaviors such as instability and bifurcation. In practical applications, one often needs to find analytical form of the governing equations involving perturbation and/or control parameters, which can directly generate stability conditions and bifurcation diagrams. Center manifold theory and normal form theory are two of the useful and important tools in the study of bifurcation and stability of nonlinear systems. They are used to transform the original problem to a simpler form before trying to solve it. However, finding a normal form for a given system of differential equations is not a simple task. For applications, it usually requires computing the explicit expressions of center manifolds and normal forms in terms of the coefficients of the original nonlinear system, which is a very involved procedure and symbolic computation must be employed. In particular, to prove the existence of multiple limit cycles in high dimensional dynamical systems which usually contain a number of free parameters, center manifold and normal form computations are even more computationally demanding. Therefore, development of fundamental theory and efficient computational methods for center manifold and normal form are necessary in order to analyze higher dimensional, nonlinear problems. The objectives of this project include developing theory and efficient method for computing center manifold and normal form of general nonlinear dynamical systems, developing user-friendly automated Maple programs, and application to study bifurcation of multiple limit cycles in biological and engineering systems.