  The Lorenz system is a milestone model of nonlinear dynamic systems. However,
we report in this Letter that important information of the global solutions in
the parameter space may still miss: there is a series of cascade solutions in
certain regions of the parameter space that has not been found before. We
denote them as Si, where i=1,2,3,..., can tend to be infinite. The solution S1
is the fundamental solution having been studied intensively, the second one has
also been reported and studied partially before. We conduct a detailed study on
other solutions to reveal their special self-similar features, which enable us
to find that solutions of i=1,3,5,..., and solutions with i=2,4,6,..., have
different symmetry, but in each series all of the solutions are qualitatively
similar after properly rescaled. The phase diagram of cascade solutions is
shown in the parameter space. Odd and even solutions may coexist in certain
parameter regions, in which case their basins of attraction show fractal
geometry. With high-order solutions, the Lorenz system is shown analog to an
excitable system. An ordinary differential equation existing a series of
infinite cascade solutions has never been reported before.
