A note on “Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system” in [Appl. Math. Comput. 291 (2016) 323–339] by Amin Zarei and Saeed Tavakoli

A note on “Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic... In the recent paper entitled “Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system” in [Appl. Math. Comput. 291 (2016) 323–339] by Amin Zarei and Saeed Tavakoli, they proposed the following new four-dimensional (4-D) quadratic autonomous hyper-chaotic system: x1˙=a(x2−x1), x2˙=bx1−x2+ex4−x1x3, x3˙=−cx3+x1x2+x12, x4˙=−dx2, which generates double-wing chaotic and hyper-chaotic attractors with only one equilibrium point. Combining theoretical analysis and numerical simulations, they investigated some dynamical properties of that system like Lyapunov exponent spectrum, bifurcation diagram, phase portrait, Hopf bifurcation, etc. In particular, they formulated a conclusion that the system has the ellipsoidal ultimate bound by employing the method presented in the paper entitled “Ultimate bound estimation of a class of high dimensional quadratic autonomous dynamical systems” [Int. J. Bifurc. Chaos, 21(09) (2011), 2679–2694] by P. Wang et al. However, by means of detailed theoretical analysis, we show that both the conclusion itself and the derivation of its proof in [Appl. Math. Comput. 291 (2016) 323–339] are erroneous. Furthermore, we point out that the method adopted for studying the ultimate bound of that system is not applicable at all. Therefore, the ultimate bound estimation of that system needs further studying in future work. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Computation Elsevier

A note on “Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system” in [Appl. Math. Comput. 291 (2016) 323–339] by Amin Zarei and Saeed Tavakoli

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Publisher
Elsevier
Copyright
Copyright © 2018 Elsevier Inc.
ISSN
0096-3003
eISSN
1873-5649
D.O.I.
10.1016/j.amc.2018.01.027
Publisher site
See Article on Publisher Site

Abstract

In the recent paper entitled “Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system” in [Appl. Math. Comput. 291 (2016) 323–339] by Amin Zarei and Saeed Tavakoli, they proposed the following new four-dimensional (4-D) quadratic autonomous hyper-chaotic system: x1˙=a(x2−x1), x2˙=bx1−x2+ex4−x1x3, x3˙=−cx3+x1x2+x12, x4˙=−dx2, which generates double-wing chaotic and hyper-chaotic attractors with only one equilibrium point. Combining theoretical analysis and numerical simulations, they investigated some dynamical properties of that system like Lyapunov exponent spectrum, bifurcation diagram, phase portrait, Hopf bifurcation, etc. In particular, they formulated a conclusion that the system has the ellipsoidal ultimate bound by employing the method presented in the paper entitled “Ultimate bound estimation of a class of high dimensional quadratic autonomous dynamical systems” [Int. J. Bifurc. Chaos, 21(09) (2011), 2679–2694] by P. Wang et al. However, by means of detailed theoretical analysis, we show that both the conclusion itself and the derivation of its proof in [Appl. Math. Comput. 291 (2016) 323–339] are erroneous. Furthermore, we point out that the method adopted for studying the ultimate bound of that system is not applicable at all. Therefore, the ultimate bound estimation of that system needs further studying in future work.

Journal

Applied Mathematics and ComputationElsevier

Published: Jul 15, 2018

References

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