On singular orbits and a given conjecture for a 3D Lorenz-like system

Abstract

We revisit a 3D chaotic system in Dias et al. (Nonlinear Anal Real World Appl 11(5): 3491–3500, 2010) and mainly consider its singular orbits not yet investigated: homoclinic and heteroclinic orbits and singularly degenerate heteroclinic cycles. We first consider the existence of homoclinic and heteroclinic orbits. Our results, one of which shows the existence of two heteroclinic orbits for $$c \ge 2a > 0$$ and $$b > 0$$ , not only further supplement the ones obtained in this literature, but also give something new to theoretically helpfully understand the occurrence of chaos. Further, numerical simulations show that this system has not only two heteroclinic orbits for $$a \le c 0$$ or $$a > c > 0$$ and some $$ b_{0} \in (0, \frac{a+c}{a-c})$$ , but also chaotic attractor when heteroclinic orbits disappear. Then, by utilizing a known conclusion, we demonstrate the existence of singularly degenerate heteroclinic cycles in this system. Combining analytical and numerical techniques, it is shown that for the parameter value $$c = 0$$ the system presents an infinite set of singularly degenerate heteroclinic cycles, which completely solves a conjecture presented in the above literature for the existence of infinitely many singularly degenerate heteroclinic cycles in the system.