RECONSTRUCTION OF SOME NONLINEAR NONAUTONOMOUS SYSTEMS BY USE OF SCALAR TIME SERIES - Gorodetskyi, V.G. (Computer Science and Applied Mathematics, 2016)
The problem of obtaining mathematical models for systems with external excitation is of interest both from a theoretical and practical point of view. Such systems are widely used in physics, engineering, biology and other applications. The article deals with the special case of the inverse problem - reconstruction of nonlinear system of ordinary differential equations with external excitation by use of a single observed variable represented by a scalar time series. We try to establish a strict mathematical conditions for identification of the systems with unknown periodic excitation. Such problem may arise especially in the case when the system is acting in the regime of deterministic chaos. This study proposes the formulation and proof of the theorems, which establish the necessary and sufficient conditions for periodicity of the unknown outer excitation. They also allow to find unknown period of the outer excitation. The identification of other unknown parameters of the system thus becomes less complicated. The possibility of practical application of proposed theory is illustrated by a numerical examples. The possible difficulties that may arise in the implementation of the algorithm are identified. One of them is the bad condition number, that may appear while forming the matrix to solve inverse problem. This is clearly demonstrated by numerical examples and graphs. To create algebraic system for solving the inverse problem one must use numerical differentiation. This operation also may be the reason of increasing errors because of numerical noise. That is why the researcher must use special type of numerical differentiation, filtering, smoothing, etc. The proved theorems can be useful for building an algorithm identifying some systems with external excitation on the basis of data on a single observable variable.