Reconstruction Of The Ordinary Differential System Based On The Discrete Normal Vectors

There are a lot of differential dynamic systems applied in many fields. Most of researches focus on the properties of the phase trajectories, but the study on it's inverse problem, i.e., how to reconstruct a differential dynamic system by some given discrete data points, is still relatively few. The expressions of the phase trajectories of differential dynamic systems contain exponential matrix, so it is difficult to compute. Difference method is a basic method of linearization methods to deal with the nonlinear terms. The difference method need to parameterize the data points, but the fitting result is sensitive to the parameterization. This thesis discuss the reconstruction of ordinary differential systems based on the discrete normal vectors.(1) By analyzing the homogeneous linear ordinary differential system which is the simplest differential dynamic system, it is difficult to construct the optimization model directly. By the orthogonal properties of the tangent vectors and the normal vectors and the least squares method, we develop an optimization model and the corresponding algorithm 1. Then we discuss the uniqueness of the solution of the model. The numerical examples demonstrate the effectiveness of the algorithms with different type normal vectors. (2) The singularities of the homogeneous linear ordinary differential systems all are origin, but the singularities of the curves are not origin in practical applications. However, the differential dynamic system don't satisfy translational invariant, so this leads to the reconstruction problem of non-homogeneous linear ordinary differential systems. Similarly, we develop the optimization model and the corresponding algorithm 2 and discuss the uniqueness of the solution of the model. The numerical examples demonstrate the effectiveness of the algorithms with different normal vectors. (3) There are many nonlinear differential dynamic systems in practical applications, we also discuss the reconstruction problem of those systems in the last chapter and develop the optimization model and the corresponding algorithm 3.The method of this thesis have two main advantages:(1) the discrete data points can be fitted by a curve possessing dynamic properties; (2) this method avoids the parameterization of the data points.