Self-similar attractor sets of the Lorenz model in parameter space

Although the Lorenz model has been extensively studied for over half a century, detailed exploration of its parameter space can still yield new discoveries. By decreasing the parameter b of the model, we find infinite attracting sets, of which only two have been previously revealed. We denote each set as Si and demonstrate that they can be divided into two series: the odd-indexed (i = 1, 3, ...) and even-indexed (i = 2, 4, ...) series. The bifurcation diagrams of the attracting sets in each series exhibit structural similarity. The well-known and thoroughly studied attractors are from S-1 and S-2, i.e., the leading attracting sets in the odd-indexed and even-indexed series, respectively; they occupy most of the parameter space of interest. The newly uncovered attracting sets reside in the parameter region with very small values of b, and their trajectories resemble those of an excitable system. Adjacent attracting sets can coexist within a parameter range, and in such cases, their basins of attraction exhibit a fractal geometry.