Comment on "Existence of heteroclinic and homoclinic orbits in two different chaotic dynamical systems" [Appl. Math. Comput. 218 (2012) 11859-11870]

Abstract In the commented paper, the authors claim to have proved the existence of heteroclinic and homoclinic orbits of Silnikov type in two-Lorenz like systems, the so-called Lu and Zhou systems. According to them, they have analytically demonstrated that both systems exhibit Smale horseshoe chaos. Unfortunately, we show that the results they obtain are incorrect. In the proof, they use the undetermined coefficient method, introduced by Zhou et al. in [Chen’s attractor exists, Int. J. Bifurcation Chaos 14 (2004) 3167–3178], a paper that presents very serious shortcomings. However, it has been cited dozens of times and its erroneous method has been copied in lots of papers, including the commented paper where a misuse of a time-reversibility property leads the authors to use an odd (even) expression for the first component of the heteroclinic (homoclinic) connection. It is evident that this odd (even) expression cannot represent the first component of a Silnikov heteroclinic (homoclinic) connection, an orbit which is necessarily non-symmetric. Consequently, all their results, stated in Theorems 3–5, are invalid.