P. Kuptsov
Publications:
Kuptsov P. V., Kuptsova A. V., Stankevich N. V.
Artificial Neural Network as a Universal Model of Nonlinear Dynamical Systems
2021, Vol. 17, no. 1, pp. 521
Abstract
We suggest a universal map capable of recovering the behavior of a wide range of dynamical
systems given by ODEs. The map is built as an artificial neural network whose weights encode
a modeled system. We assume that ODEs are known and prepare training datasets using the
equations directly without computing numerical time series. Parameter variations are taken into
account in the course of training so that the network model captures bifurcation scenarios of the
modeled system. The theoretical benefit from this approach is that the universal model admits
applying common mathematical methods without needing to develop a unique theory for each
particular dynamical equations. From the practical point of view the developed method can be
considered as an alternative numerical method for solving dynamical ODEs suitable for running
on contemporary neural network specific hardware. We consider the Lorenz system, the Rцssler
system and also the Hindmarch – Rose model. For these three examples the network model
is created and its dynamics is compared with ordinary numerical solutions. A high similarity
is observed for visual images of attractors, power spectra, bifurcation diagrams and Lyapunov
exponents.

Kuptsov P. V., Kuznetsov S. P.
Transition to a synchronous chaos regime in a system of coupled nonautonomous oscillators presented in terms of amplitude equations
2006, Vol. 2, No. 3, pp. 307331
Abstract
Amplitude equations are obtained for a system of two coupled van der Pol oscillators that has been recently suggested as a simple system with hyperbolic chaotic attractor allowing physical realization. We demonstrate that an approximate model based on the amplitude equations preserves basic features of a hyperbolic dynamics of the initial system. For two coupled amplitude equations models having the hyperbolic attractors a transition to synchronous chaos is studied. Phenomena typically accompanying this transition, as riddling and bubbling, are shown to manifest themselves in a specific way and can be observed only in a small vicinity of a critical point. Also, a structure of manydimensional attractor of the system is described in a region below the synchronization point.
