# Research on the Stability Model in Discrete Dynamical Systems with the Lorenz Attractor and the Kropotov–Pakhomov Neural Network

## TL;DR

This paper shows how a modified neural network can emulate the chaotic behavior of the Lorenz attractor and share similar stability properties.

## Contribution

The study introduces a modified Kropotov–Pakhomov neural network as a discrete model of the Lorenz attractor with equivalent stability conditions.

## Key findings

- The MRNN can reproduce the hierarchy of modes of the peak map derived from the Lorenz system.
- The stability conditions of the MRNN are equivalent to those of the Lorenz attractor through the map multiplier and monodromy matrix spectral radius.
- Numerical results confirm the correspondence between the Lorenz model and the MRNN parameters.

## Abstract

This paper explores the dynamic analogy between the discrete Lorenzian attractor and a modified Kropotov–Pakhomov neural network (MRNN). A one-dimensional peak map is used to extract the successive maxima of the Lorenzian system and preserve the basic properties of the chaotic flow. The MRNN, governed by the Bogdanov–Hebb learning rule with dissipative feedback, is formulated as a discrete nonlinear operator whose parameters can reproduce the same hierarchy of modes as the peak map. It is theoretically shown that the map multiplier and the spectral radius of the monodromy matrix of the MRNN provide equivalent stability conditions. Numerical diagrams confirm the correspondence between the control parameters of the Lorenz model and the network parameters. The results establish the MRNN as a neural emulator of the Lorenz attractor and offer an analysis of self-organization and stability in adaptive neural systems.