Coexistence of multiscroll chaotic attractors in two coupled inertial Hopfield neurons: numerical simulations and experiment

In this contribution, we consider the dynamics of a pair of coupled inertial neurons with hyperbolic tangent activation function. The two inertial neurons are coupled by adding to each one's amplitude a perturbation proportional to the other one. The model is governed by a fourth-order autonomous system with hyperbolic tangent nonlinearities. The analysis of the coupled system yields nine equilibrium points some of which experience Hopf type bifurcation. When adjusting the coupling coefficients, striking nonlinear patterns are disclosed such as the coexistence of numerous bifurcation branches, merging crisis, multiple Hopf bifurcations, coexisting self-excited motions (e.g. two coexisting double-scroll chaos, four coexisting period-n cycles, four coexisting single-scroll chaos), and four-scroll chaotic attractors. These latter features are diagnosed with the help of classic numerical tools (e.g. 1D and 2D maximum Lyapunov exponent diagrams, 1D bifurcation diagrams of local peaks of variables, frequency spectrum plots, phase space trajectory plots, and attraction basins). The analogue electronic circuit design of the coupled inertial neurons system is carried out and simulated in PSpice to verify diverse types of features reported during the theoretical study. One of main achievements of the present article is that the coupling of inertial neurons can be regarded as an alternative scheme to obtain multiscroll chaotic signals.

Automated summary

This study explores the dynamics of a pair of coupled inertial neurons with hyperbolic tangent activation function, revealing various nonlinear patterns and features such as coexisting bifurcation branches, merging crisis, and coexisting self-excited motions. Additionally, an analogue electronic circuit design is conducted to verify the diverse types of features reported during the theoretical study.