Experimental detection of Hopf bifurcation in two-dimensional dynamical systems

Abstract

In this work, we propose a strategy based on an analog active network to detect Hopf bifurcations in two–dimensional dynamical systems described by ordinary differential equations. With the proposed strategy, two parameters of the nonlinear system are established in the analog active network by using external controllable voltage levels in order to explore the dynamical evolution of the system in a fast, easy, and accessible way, making our approach a powerful tool to detect Hopf bifurcations in a two-dimensional space. To demonstrate the proposed strategy’s potential and functionality, we electronically implement the kinetics of a reaction-diffusion model proposed by Barrio et al. in 1999 [1], called BVAM model. Hopf bifurcations are detected by following the changes in the system, going from stationary to periodic solutions when varying the control parameters. Local linear stability analysis is performed to show the quantitative agreement between analytical and experimental bifurcations. We additionally found that global effects are detected by the experimental approach, which cannot be predicted from a local analysis. The proposed strategy opens the way to use analog active networks to detect bifurcations in dynamical systems experimentally.