New Chaotic Regimes in the Lorenz and Chen Systems

has been much studied for positive values of its parameters (a, b, c) and has chaotic solutions for typical parameters of (35, 3, 28). Recently, Algaba et al. [2013] have shown that the Chen system has only two independent parameters a/c and b/c and that it is identical to the reversed-time Lorenz system for c > 0 and to the forward-time Lorenz system for c < 0 in the parameter plane ρ + σ = −1. Hence it follows that the Lorenz system has a strange repeller for certain negative values of its parameters corresponding to the strange attractor in the Chen system with certain positive parameters. Furthermore, Algaba et al. showed that for c < 0, Chen’s attractor exists if the Lorenz attractor exists in this unusual range of its parameters. In particular, for c = −1 the two equations are identical provided σ = a, ρ = c − a, and β = b. The purpose of this paper is to show that such attractors do exist and to determine their properties and the conditions under which they occur.