Bounds of the hyper-chaotic Lorenz–Stenflo system

Abstract 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 , d and a cylindrical bound for a  > 0, b  > 1, c  > 0, d  > 0 are derived. Numerical results show the effectiveness and advantage of our methods.