Multiple Switching and Bifurcations of In-phase and Anti-phase Periodic Orbits to Chaos Coexistence in a Delayed Half-center CPG Oscillator

Abstract

In this study, we investigate complex dynamical behaviors of a delayed HCO (half-center oscillator) neural system consisted of two inertial neurons. The neural system proposes two types of periodic orbits with 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 chaos attractor by employing the quasi-periodic orbit bifurcation. Further, the chaos attractor breaks and bifurcates into a pair of symmetry multiple-periodic orbits, which evolves into a pair of symmetry chaos attractors by the 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 the increasing of the variable bifurcation parameter τ. In the presented paper, we just exhibit 14 attractors’ coexistence induced by the multiple bifurcation routes, which includes periodic orbits, quasi-periodic orbits, chaos attractors, and multiple-periodic orbits.

Dynamical complexity of a fractional‐order neural network with nonidentical delays: Stability and bifurcation curves

“…(H2) There exists positive real roots ωk (k = 1, 2, … ) for Equation (12). From Equation ( 12), the values of ω can be acquired, then the bifurcation point τ0 of FONN ( 2) with 𝜎 = 0 can be obtained.…”

“…As a consequence, various time delays should be included in the model to provide a more realistic representation. What is exciting is that many scholars are currently researching time-delay NNs [10][11][12]. Song et al [13] proposed a type of DHCO neural model that shows more chaotic attractors and their coexistence phenomena, which provides more possibilities and means for engineering applications of robot motion control.…”