Dynamical analysis, stabilization and discretization of a chaotic fractional-order GLV model

In this paper, the fractional-order generalized Lotka-Volterra (GLV) model and its discretization are investigated qualitatively. A sufficient condition for existence and uniqueness of the solution of the proposed system is shown. Analytical conditions of the stability of the system's three non-negative steady states are proved. The conditions of the existence of Hopf bifurcation in the fractional-order GLV system are discussed. The necessary conditions for this system to remain chaotic are obtained. Based on the stability theory of fractional-order differential systems, a new control scheme is introduced to stabilize the fractional-order GLV system to its steady states. Furthermore, the analytical conditions of stability of the discretized system are also studied. It is shown that the system's fractional parameter has effect on the stability of the discretized system which shows rich variety of dynamical analysis such as bifurcations, an attractor crisis and chaotic attractors. Numerical simulations are used to support the analytical results.