Two bifurcation routes to multiple chaotic coexistence in an inertial two-neural system with time delay

In this article, we establish an inertial two-neural system with time delay and illustrate the stable coexistence of three chaotic attractors that arise via two different bifurcation routes, i.e., the period-doubling and quasi-periodic bifurcations. So, we firstly analyze the system equilibria by nullcline curves. By the pitchfork/saddle-node bifurcation of the trivial/nontrivial equilibria, the system parameter ($c_1$, $c_2)$-plane is divided into the different regions having the different number of equilibrium. Further, the trivial and nontrivial equilibria will lose their stability and bifurcate into periodic orbits as the effect of time delay. The system has the stable coexistence of two periodic orbits near the nontrivial equilibria. For some delayed regions, the system illustrates the stability switching, i.e., the dynamic behaviors lost, retrieved, and lastly lost their stability with increase in delay. Using the Hopf–Hopf bifurcation analysis, we find a quasi-periodic orbit surrounded by the trivial equilibrium. Lastly, based on numerical simulations, such as phase portrait, Poincare section, Lyapunov exponent, and one-dimensional bifurcation diagram, we further investigate the dynamical evolution of the periodic and quasi-periodic orbits. The results show that the neural system presents the multiple stable coexistence with three chaotic attractors by the different bifurcation routes, i.e., the period-doubling and quasi-periodic bifurcations.