Bounds of the hyper-chaotic Lorenz–Stenflo system

To estimate the ultimate bound and positively invariant set for a dynamical system is an important but quite challenging task in general. This paper attempts to investigate the ultimate bounds and positively invariant sets of the hyper-chaotic Lorenz–Stenflo (L–S) system, which is based on the optimization method and the comparison principle. A family of ellipsoidal bounds for all the positive parameters values a, b, c, dand a cylindrical bound for a > 0, b > 1, c > 0, d > 0 are derived. Numerical results show the effectiveness and advantage of our methods.