Multiple switching and bifurcations of in-phase and anti-phase periodic orbits to chaotic coexistence in a delayed half-center CPG oscillator

In this study, we investigate complex dynamical behaviors of a delayed-HCO (half-center oscillator) neural system consisting of two inertial neurons. The system proposes two types of periodic orbits having in-phase and anti-phase spatiotemporal patterns that arise via the Hopf bifurcation of the trivial equilibrium and the homoclinic orbit (Homo) bifurcation of the nontrivial equilibrium. With increasing time delay, the periodic orbit translates into a quasi-periodic orbit and enters chaotic attractor by employing quasi-periodic orbit bifurcation. Further, the chaotic attractor breaks and bifurcates into a pair of symmetric multiple-periodic orbits, which evolves into a pair of symmetric chaotic attractors by period-doubling bifurcation. The delayed-HCO neural system presents multiple coexistence employing two classical bifurcation routes to chaos, i.e., the quasi-periodic orbit and period-doubling bifurcations. What is interesting is that the delayed-HCO neural system proposes seven similar sequences (maybe up to infinity) of the bifurcation routes to chaos with increase in the bifurcation parameter τ. In this paper, we describe the coexistence of 14 attractors induced by the multiple bifurcation routes, which includes periodic orbits, quasi-periodic orbits, chaotic attractors, and multiple-periodic orbits.