A true three-scroll chaotic attractor coined

Based on the method of compression and pull forming mechanism (CAP), the authors in a well-known paper proposed and analyzed the Lü-like system: \begin{document}$ \dot{x} = a(y - x) + dxz $\end{document} , \begin{document}$ \dot{y} = - xz + fy $\end{document} , \begin{document}$ \dot{z} = -ex^{2} + xy + cz $\end{document} , which was thought to display an interesting three-scroll chaotic attractors (called as Pan-A attractor) when \begin{document}$ (a, d, f, e, c) = (40, 0.5, 20, 0.65, \frac{5}{6}) $\end{document} . Unfortunately, by further analysis and Matlab simulation, we show that the Pan-A attractor found is actually a stable torus. Accordingly, we find a new true three-scroll chaotic attractor coexisting with a single saddle-node \begin{document}$ (0, 0, 0) $\end{document} for the case with \begin{document}$ (a, d, f, e, c) = (168, 0.4, 100, 0.70, 11) $\end{document} . Interestingly, the forming mechanism of singularly degenerate heteroclinic cycles of that system is bidirectional, rather than unilateral in the case of most other Lorenz-like systems. This further motivates us to revisit in detail its other complicated dynamical behaviors, i.e., the ultimate bound sets, the globally exponentially attractive sets, Hopf bifurcation, limit cycles coexisting attractors and so on. Numerical simulations not only are consistent with the results of theoretical analysis, but also illustrate that collapse of infinitely many singularly degenerate heteroclinic cycles and explosions of normally hyperbolic stable nodes or foci generate the aforementioned three-scroll attractor. In particular, four or two unstable limit cycles coexisting one chaotic attractor, the saddle \begin{document}$ E_{0} $\end{document} and the stable \begin{document}$ E_{\pm} $\end{document} are located in two globally exponentially attractive sets. These results together indicate that this system deserves further exploration in chaos-based applications.