Modeling of Chaotic Processes by Means of Antisymmetric Neural ODEs

Abstract

The main goal of this work is to construct an algorithm for modeling chaotic processes using special neural ODEs with antisymmetric matrices (antisymmetric neural ODEs) and power activation functions (PAFs). The central part of this algorithm is to design a neural ODEs architecture that would guarantee the generation of a stable limit cycle for a known time series. Then, one neuron is added to each equation of the created system until the approximating properties of this system satisfy the well-known Kolmogorov theorem on the approximation of a continuous function of many variables. In addition, as a result of such an addition of neurons, the cascade of bifurcations that allows generating a chaotic attractor from stable limit cycles is launched. We also consider the possibility of generating a homoclinic orbit whose bifurcations lead to the appearance of a chaotic attractor of another type. In conclusion, the conditions under which the found attractor adequately simulates the chaotic process are discussed. Examples are given.

Study of the Dynamics of Product Sales Process with the Help of Zolotas Model

“…However, there always remains a scientific interest in studying the behavior of trajectories of system (2.1), which goes beyond the scope prescribed by Theorem 2.1. This interest is dictated by the following question: are there such values of the parameters of system (2.1), at which the appearance of periodic trajectories (see [2,3]) in the system is possible?…”

“…Let's consider the 4th equation of of system (2.1) (this is equation (1.5)). If P 2 (0) ≥ 0, then according to the Comparison Principle [2,6]), we have −µ 0 P 3 (t)− ϵµ 0 P 2 3 (t) ≤ λP 2 (t) − µ 0 P 3 (t) − ϵµ 0 P 2 3 (t). Thus, if P 3 (0) ≤ 0, then from (1.2) it follows that solution P 3 (t) ≤ 0 has a singular point t * such that lim t→t * P 3 (t) = −∞.…”

Systems of Singular Differential Equations as the Basis for Neural Network Modeling of Chaotic Processes

“…x 0 = x(t 0 ), x 1 = x(t 1 ), ..., x N = x(t N ) (1.1) be a finite sequence (time series) of numerical values of some scalar dynamical variable x(t) measured with the constant time step ∆t in the moments t i = t 0 + i∆t; x i = x(t i ); i = 0, 1, ..., N (thus, ∆t = t N /N ) [5,6,9,11,12,16,21]. The choice of equations for a model that describes the dynamics of certain processes is a difficult task.…”

“…Having parameters n and τ , we can assume that to model a process described by time series (1.1), a certain system of differential equations has already built. (In what follows, we will assume that a recurrent neural network (RNN), which is a discrete analogue of the mentioned ODE system, was also constructed [4,6,11,16,21].) Below we will focus on two areas of research, which can be formulated in the following questions.…”

“…The final sections of the article are devoted to the development of an algorithm for determining the parameters of ODE systems for a known time series. The essence of this algorithm is that it uses a special structure of neural ODEs (antisymmetric neural ODEs), with which it is possible to generate a limit cycle [6,10,13]. After this, by selecting weight coefficients, we obtain such bifurcations of the indicated cycle that lead to the modeling of a real chaotic process.…”