Abstract
A problem of description of algebraic invariants for a linear control system that determine its structure is considered. With the help of these invariants, the equivalence problem of two linear time-invariant control systems with respect to actions of some linear groups on the spaces of inputs, outputs, and states of these systems is solved. The invariants are used to establish the necessary equivalence conditions for two nonlinear systems of differential equations generalizing the well-known Hopfield neural network model. Finally, these conditions are applied to establish the adequacy of two neural network models designed to describe the behavior of a real dynamic process given by two different sets of time series.
Highlights
In recent decades, researchers have paid much attention to chaotic behavior in many fields, such as meteorology, medicine, economics, signal processing, traffic flow, and many others [16,34,36,49]
Researchers have paid much attention to chaotic behavior in many fields, such as meteorology, medicine, economics, signal processing, traffic flow, and many others [16,34,36,49]. They developed many models describing chaotic time series in order to predict the behavior of these time series
Researchers have found that it is a difficult problem to forecast chaotic time series, which are the evolution of chaotic systems, with the use of traditional time series forecasting methods [16, 36]
Summary
Researchers have paid much attention to chaotic behavior in many fields, such as meteorology, medicine, economics, signal processing, traffic flow, and many others [16,34,36,49]. We assume that using suitable methods (based on Theorem 1.1) the system of ordinary autonomous differential equations, the solution x(t) ∈ Rn of which simulates process P(t) with a given accuracy, was reconstructed [7] We assume that this system (with the known vector of initial values xT (0) = (x10, ..., xn0)) has the following form:. Mathematical tools have been developed to test the adequacy, based on an algebraic theory of invariants The idea of such verification is based on the following well-known fact: with arbitrary observations of a dynamic process, there are always functions that are independent of the methods of observations, but depend on an internal structure that determines the behavior of the process. These parameters are determined by the structure of the time series (1.1) and choice of approximating functions in the right-hand sides of the obtained system of differential equations
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