Coexisting Modes of Oscillations and Grid-Scroll Chaos in Two Coupled Hopfield-Type Inertial Neurons: Theoretical Study and Microcontroller-Based Experiment

In this paper, we consider the dynamics of a system consisting of two inertial neurons whose activation function is of the hyperbolic tangent type. We distinguish two cases: a homogeneous coupling where each of the two neurons has an activation function with a variable gradient; then, a heterogeneous coupling where one of the two neurons has an activation function with a variable gradient and the other an activation function with a fixed gradient. In each case, the equilibrium points of the coupled system are determined and their stability is examined according to Routh’s criterion combined with Hopf’s bifurcation theorem. Next, we examine the path that leads the Hopf bifurcation system toward grid-scroll chaos when the coupling coefficients are gradually varied. To investigate this, we mainly use the drawings of bifurcation diagrams, basins of attraction and phase portraits. For homogeneous coupling, the system presents a chaotic [Formula: see text] scroll chaotic attractor, while the attractor is of the [Formula: see text] grid-scroll type for heterogeneous coupling. The route to chaos in each case is also marked by the presence of several parameter zones where the system exhibits multistable behavior. To verify these theoretical analysis results, an experimental study is carried out using an Arduino card.

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