Templates and subtemplates of Rössler attractors from a bifurcation diagram (Q2816806)

The bifurcation diagram of the following Rössler system parameterized by \(\alpha\) is studied in [\textit{J. C. Sprott} and \textit{C. Li}, ``Asymmetric bistability in the Rössler system'', Int. J. Bifurcation Chaos (to appear)]; NEWLINE\[NEWLINE\begin{cases} \dot{x} \, = \, -y-z \\ \dot{y} \, = \, x+ay \\ \dot{z} \, = \, b+z(x-c) \end{cases}\qquad {\text{with}} \quad \begin{cases} a \, = \, 0.2+0.09\alpha \\ b \, = \, 0.2 - 0.06 \alpha \\ c \, = \, 5.7-1.18 \alpha \end{cases} NEWLINE\]NEWLINE The present paper states that in previous works it has been shown that when \(\alpha \, = \, 1\) two attractors coexist in the phase space and the bifurcation diagram for \(\alpha \in (-2,1.5)\) has been given.NEWLINENEWLINEWith these results in mind the author chooses eight representative values of \(\alpha\) where one or two attractors are solutions of the system, namely \(\alpha =-0.25\), \(\alpha =0.5\), \(\alpha =0.78\), \(\alpha =0.86\), \(\alpha =1\) (twice), \(\alpha =1.135\) and \(\alpha =1.22\). Then the author uses the topological characterization method to build a template for each of the eight attractors using topological properties of periodic orbits. Quoting the paper, a template is a compact branched two-manifold with boundary and smooth expansive semiflow, endowing the template with an expanding semiflow, and the gluing maps between charts must respect the semiflow and at linearly on the edges.NEWLINENEWLINEThen the author shows that the eight templates are subtemplates of a unique template. Here a template is a tool to describe one attractor and a set of neighbor attractors (in the parameter space). This description allows the illustration of the chaotic mechanism of the system.