The dynamics of NE9 and NE8 systems, and predicting chaos

Abstract

We considered the 3-dimensional systems of NE8 and NE9, which depend on a single non negative parameter. We proved that in both systems, when the parameter is set to zero, have a line of equilibria in the z-axis. Moreover, the eigenvalues and the behavior of the eigenvectors indicate that the positive z-axis is an attractor. On the other hand, when the parameter is positive, a periodic orbit in the neighborhood of the origin appeared. The periodic orbit was revealed using the averaging method and represented analytically. However, in the NE9 system, nested tori appeared but were demolished at a range of initial values. Moreover, when the parameter is small and the initial radius is large, a limit set and a chaotic attractor appeared. Furthermore, the existence of a chaotic attractor was shown numerically in the NE9 system. On the other hand, in the NE8 system, we proved analytically and showed numerically that the periodic orbit is an attractor and that several attractors appeared at different values of the parameter.