The global dynamics of a new fractional-order chaotic system

This paper investigates the global dynamics of a new 3-dimensional fractional-order (FO) system that presents just cross-product nonlinearities. Firstly, the FO forced Lorenz-84 system is introduced and the stability of its equilibrium points, as well as the chaos control for their stabilization, are addressed. Secondly, dynamical behavior is further analyzed and bifurcation diagrams, phase portraits, and largest Lyapunov exponent (LE) are discussed. Then, the global Mittag-Leffler attractive sets (MLASs) and Mittag-Leffler positive invariant sets (MLPISs) of the FO forced Lorenz-84 system are presented. Finally, the Hamilton energy function (HEF) of the Lorenz-84 system is calculated by using the Helmholtz theorem. The calculation of the Hamilton energy has an essential role on the estimation of chaos in dynamical systems, the guidance of orbits, and stability. In fact, any control action on the dynamical system completely changes the HEF. Numerical simulations are presented for illustrating the theoretical findings.