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Methods for bifurcation and recurrent analysis of nonlinear dynamical systems on memristive circuit example

The subject of this study is the analysis algorithms for studying the behavior of the computer models of electronic circuits with advanced nonlinear elements. The paper gives an experimental comparison of four methods of obtaining two-parametric bifurcation diagrams: histogram of maximal values method, kernel density estimation method, sliding window technique and algorithm based on clusterization of recurrence density plot. The simple memristive circuit with chaotic multi-scroll attractor was chosen as an example for experimental part of the study. The following main results were obtained. Two new algorithms of chaotic system analysis were developed, combining the methods of bifurcation and recurrent analysis with mathematical statistics. New technique for creation of two-parametric high resolution bifurcation diagrams is proposed. This technique is based on sliding average window approach. We also propose the clustering method for plotting precise dynamical maps. The comparative analysis of proposed techniques is described, including the comparison with existing approaches. The study of computational efficiency is performed for integration numerical methods applied for the synthesis of discrete models of the circuit. By comparison of efficiency plots the optimal ordinary differential equation (ODE) solver for computer experiments with discrete models was chosen. The efficiency analysis has shown that the best ODE solver for simple memristive circuit simulation is the semi-implicit extrapolation algorithm of order eight. It is experimentally shown that sliding window average method is the most precise approach for creation of two-parametric bifurcation diagrams. The practical relevance of the obtained results is as follows. New methods of nonlinear systems analysis are applicable for creation of high precision simulation tools for circuits with advanced nonlinear elements. The applications are not limited to the electronic circuits simulation only but also include the possibility to locate hidden attractors in chaotic dynamical systems.