On the dynamics of five- and six-dimensional Lorenz models

Abstract

In this paper we report on generalized Lorenz models. Five- and six-dimensional Lorenz models are investigated, which are obtained by considering respectively two and three additional Fourier modes in addition to the modes included in the derivation of the classical three-dimensional Lorenz model. Parameter planes, bifurcation diagrams, and attractors in the phase-space are used, in order to investigate the influence of the additional Fourier modes on solutions, when compared with the solutions for the classical Lorenz model. It is shown that for parameters σ and b kept fixed, a larger parameter r results for the onset of chaos in five-and six-dimensional Lorenz models. Also it is shown that the shape of bifurcation diagrams, periodic, and chaotic attractors is preserved in both generalized Lorenz models. Additionally, it is shown that hyperchaos is observed only in the six-dimensional Lorenz model, at least in the parameter ranges here investigated.