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, Poincaré 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.
This is a preview of subscription content, log in via an institution to check access.
References
Chang, P., Kim, D.: Introduction and synchronization of a five-term chaotic system with an absolute-value term. Nonlinear Dyn. 73(1–2), 311–323 (2013)
Wiggins, S.: Global Bifurcations and Chaos: Analytical Methods. Springer, New York (1988)
Hastings, S.P., Troy, W.C.: A shooting approach to the Lorenz equations. Bull. Am. Math. Soc. 27(2), 298–298 (1992)
Hassard, B.D., Zhang, J.: Existence of a homoclinic orbit of the Lorenz system by precise shooting. SIAM J. Math. Anal. 25(1), 179–196 (1994)
Hastings, S.P., Troy, W.C.: A shooting approach to chaos in the Lorenz equations. J. Differ. Equ. 127(6), 41–53 (1996)
Chen, X.: Lorenz equations. Part I: existence and nonexistence of homoclinic orbits. SIAM J. Math. Anal. 27(4), 1057–1069 (1996)
Bao, J., Yang, Q.: Complex dynamics in the stretch-twistfold flow. Nonlinear Dyn. 61(4), 773–781 (2010)
Li, T., Chen, G., Chen, G.: On homoclinic and heteroclinic orbits of Chen’s system. Int. J. Bifurc. Chaos 16(10), 3035–3041 (2006)
Tigan, G., Constantinescu, D.: Heteroclinic orbits in the \(T\) and the Lü system. Chaos Solitons Fractals 42(1), 20–23 (2009)
Liu, Y., Yang, Q.: Dynamics of the Lü system on the invariant algebraic surface and at infinity. Int. J. Bifur. Chaos 21(9), 2559–2582 (2011)
Li, X., Ou, Q.: Dynamics of a new Lorenz-like chaotic system. Nonlinear Dyn. 65(3), 255–270 (2011)
Li, X., Wang, H.: Homoclinic and heteroclinic orbits and bifurcations of a new Lorenz-type system. Int. J. Bifur. Chaos 21(9), 2695–2712 (2011)
Liu, Y., Pang, W.: Dynamics of the general Lorenz family. Nonlinear Dyn. 67(2), 1595–1611 (2012)
Li, X., Wang, P.: Hopf bifurcation and heteroclinic orbit in a 3D autonomous chaotic system. Nonlinear Dyn. 73(1–2), 621–632 (2013)
Chen, Y., Yang, Q.: Dynamics of a hyperchaotic Lorenz-type system. Nonlinear Dyn. 77(3), 569–581 (2014)
Wang, H., Li, X.: More dynamical properties revealed from a 3D Lorenz-like system. Int. J. Bifurc. Chaos 24(10), 29 (2014). doi:10.1142/S0218127414501338
Acknowledgments
This work is partly supported by NSF of China (Grant: 61473340, 10771094), the Postgraduate Innovation Project of Jiangsu Province (Grant: \(\hbox {KYZZ}_{-}0361\)) and the NSF of Yangzhou University.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, H., Li, X. 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. Nonlinear Dyn 81, 1017–1019 (2015). https://doi.org/10.1007/s11071-015-1990-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-015-1990-8