Dynamical analysis and adaptive synchronization of a new 6D hyperchaotic system with the cosine function

This paper presents a novel 6D dynamic system derived from modified second-type 3D Lorenz equations using state feed- back control. While these original 3D equations are structurally simpler than the classical Lorenz equations, they generate more topologically complex attractors with a distinctive two-winged butterfly structure. The proposed system is the most compact of its kind in the literature, containing only 11 terms: two cross-product nonlinearities, two piecewise linear functions, one cosine function, five linear terms, and one constant. The newly developed 6D hyper- chaotic system exhibits rich dynamic properties, including hidden attractors and dissipative behavior. A detailed dynamic analysis has identified two unstable hyperbolic equilibrium points, indicating the potential for self-exciting attractors. Additionally, bifurcation diagrams were constructed, Lyapunov exponents were computed, and the maximum Kaplan-Yorke dimension DKY = 3.23 was obtained at parameter value a = 0.5, revealing the high complexity of the hyperchaotic dynamics. Furthermore, multistability and offset boosting control were examined to gain deeper insights into the system’s behavior. Finally, synchronization between two identical 6D hyperchaotic systems was successfully achieved using an adaptive control method.