A note on “Introduction and synchronization of a five-term chaotic system with an absolute-value term” in [Nonlinear Dyn. 73 (2013) 311–323] by Pyung Hun Chang and Dongwon Kim

Abstract

In the paper entitled “Introduction and synchronization of a five-term chaotic system with an absolute-value term” in [Nonlinear Dyn. 73 (2013) 311–323], Pyung Hun Chang and Dongwon Kim proposed the following 3D chaotic system \(\dot{x}= a(y - x),\, \dot{y}= xz,\, \dot{z}= b|y| - y^{2}\). Combining theoretical analysis with numerical technique, they studied its dynamics, including the equilibria and their stability, Lyapunov exponents, Kaplan–Yorke dimension, frequency spectrum, Poincare maps, bifurcation diagrams and synchronization. In particular, the authors formulated a conclusion that the system has two and only two heteroclinic orbits to \(S_{0}=(0, 0, 0)\) and \(S_{\pm }=(\pm b, \pm b, 0)\) when \(b\ge 2a >0\). However, by means of detailed analysis and numerical simulations, we show that both the conclusion itself and the derivation of its proof are erroneous. Furthermore, the conclusion contradicts Lemma 3.2 in the commented paper. Therefore, the conclusion in that paper is wrong.