A hidden chaotic attractor in the classical Lorenz system

Abstract

Chaotic attractors in the classical Lorenz system have long been known as self-excited attractors. This paper, for the first time, reveals a novel hidden chaotic attractor in the classical Lorenz system. Either a self-excited or a hidden chaotic attractor is now possible in the classical Lorenz system depending on values of both system parameters and initial conditions. A systematically exhaustive computer search is employed to directly search for the hidden chaotic attractor with elegant values of both system parameters and initial conditions. Time series of trajectories, Lyapunov exponents, and bifurcations of the hidden chaotic attractor are reported. Basins of attraction of individual equilibria are depicted to verify that the hidden chaotic attractor is found. Dynamic regions of attractors are illustrated to reveal seamless connections between self-excited and hidden chaotic attractors in the classical Lorenz system with wide ranges of parameters.