Solving stiff ordinary differential equations and partial differential equations using analog computing based on cellular neural networks

Abstract

Setting analog cellular computers based on cellular neural networks systems (CNNs) to change the way analog signals are processed is a revolutionary idea and a proof as well of the high importance devoted to the analog simulation methods. We provide an in-depth description of the concept exploiting analog computing based on the CNN paradigm to solve nonlinear and highly stiff ordinary differential equations (ODEs) and partial differential equations (PDEs). We appply our method to the analysis of the dynamics of two systems modeled by complex and stiff equations. The first system consists of three coupled Rössler oscillators in a Master-Slave-Auxiliary configuration. The capabilities of this coupled system to exhibit regular and chaotic dynamics have been demonstrated so far. The synchronization modes of the coupled system can be exploited in chaotic secure communications. The second system is the Burgers' equation which is a well-known classical model for analyzing macroscopic traffic flow motions/ scenarios. As a proof of concept of the proposed approach, the results obtained in this paper are compared with the results available in the relevant literature (benchmarking) and, the proposed concept is validated by a very good agreement obtained. The computation based CNNs paradigm is advantageous as it provides accurate and ultra-fast solutions of very complex ODEs and PDEs and performs real-time computing.