Global dynamical analysis of the integer and fractional 4D hyperchaotic Rabinovich system

In this paper, the dynamical behavior of integer and fractional 4D hyperchaotic Rabinovich system is studied. We use the Lagrange coefficient method to solve an optimization problem analytically so that find an accurate ultimate bound set (UBS) for the 4D hyperchaotic Rabinovich system. Furthermore, bifurcation diagrams, Lyapunov exponents, global attractive sets (GASs), and positive invariant sets (PISs) of the fractional-order system are studied. Finally, using the Mittag-Leffler function and Lyapunov function method, the Mittag-Leffler GAS and Mittag-Leffler PIS of the proposed system are estimated. Our investigations indicate that there is a close relationship between changing the parameters and the dynamical behavior of the system, Hamilton energy consumption and changing the bound of the variables. The corresponding boundedness is numerically verified to show the effectiveness of the theoretical analysis.