Global Dynamical Analysis of a 4D-Chaotic System with Application in Locating the Hidden Attractors and Synchronization

The aim of this paper is to study the dynamical analysis of an integer and fractional-order 4D-chaotic system. Based on the Lagrange multiplier method, an optimization problem has been solved analytically to calculate a precise ultimate bound set of the 4D-chaotic system. The Hamilton energy function of the 4D-chaotic system is investigated. Interestingly, the idea of bound set has been utilized to locate the hidden attractors of a particular system. The findings from bound sets have been applied to achieve the synchronization between two chaotic systems. Furthermore, this paper introduces a novel 4D-chaotic system of fractional-order. For varying the parameter values, the dynamical behavior of the fractional system is analyzed and Hamilton energy function, Lyapunov exponents, time series and bifurcation diagrams are also discussed. Finally, Mittag-Leffler globally attractive sets and Mittag-Leffler positively invariant sets of the fractional system are determined. The achieved bound sets and other theoretical results have been quantitatively tested by numerical simulations.