A New Mathematical Model of Dynamic Processes in Direct Current Traction Power Supply System

Abstract

A new autonomous 4D nonlinear model with two nonlinearities describing the dynamics of change of voltage and current in the contact railway electric network is offered. This model is a connection of two 2D oscillatory circuits for current and voltage in the contact electric network. In the found system for the defined values of parameters an existence of limit cycles is proved. By introduction of new variables this system can be reduced to 5D system only with one quadratic nonlinearity. The constructed model may be used for the control by voltage stability in a direct current power supply system.

Modeling of Chaotic Processes by Means of Antisymmetric Neural ODEs

“…(The case D 0 = I is not excluded.) Then, for certain values of the parameters, the homoclinic orbit will exist among the solutions of system (26). (Note that in system (26) the components h i (v i ), i = 1, ..., n, of vector h(v) can be either even or odd activation functions.)…”

“…Thus, if there are numbers k, i, j such that h k ((A − rI)G i x) − h k ((A − rI)G j x) ≡ 0, then the k-th equation of system ( 26) is a composition of only odd activation functions (linear and φ(u k ), where u k is a function of x); k ∈ {1, ..., n}; i = j. Since any of these functions separates points (see Definition 1), the k-th equation of system (26) satisfies the approximation Theorem 2. (Let c 0 = 0.…”

“…We assume that we know the dimension n of the real phase space in which the considered dynamic process x(t) = (x 1 (t), ..., x n (t)) T ∈ R n takes place [17,25,26]. We will also assume that the functions x 1 (t), ..., x n (t) are continuous and differentiable with respect to time t on the interval [0, ∞).…”

“…The use of representation (26) to approximate derivatives can be motivated by the following considerations.…”

“…It acts as a controlled noise source to ensure uninterrupted access to previous memorized images and memorizing new ones. Chaos allows the system to be always active, ridding it of the need to wake up or enter a stable state every time the input changes [26]. Many researchers agree that the best from the point of view of storing and processing information is the regime of ordered chaos [27].…”