Dynamics and control in a novel hyperchaotic system

  • A. E. MatoukEmail author


In this work, a novel hyperchaotic system is introduced. The system consists of four coupled continuous-time ordinary differential equations with three quadratic nonlinearities. Based on the center manifold and local bifurcation theorems, the existence of pitchfork bifurcation is proved at the origin equilibrium point of the proposed system. Also, the existence of Hopf bifurcation near all the equilibrium points of the system is shown. Moreover, stability analysis of the resulting periodic solutions is analyzed using Kuznetsov’s theory which determines the analytical conditions for the occurrence of supercritical (subcritical) Hopf bifurcation’s type. Numerical verifications such as Lyapunov exponents’ spectrum, Lyapunov dimension, bifurcation diagrams and the continuation software MATCONT are used to show the rich dynamics of the proposed system and to confirm the analytical results. Finally, the hyperchaotic behaviors in this system are suppressed to its three equilibrium points using a novel control method based on Lyapunov stability approach.


Novel hyperchaotic system Pitchfork bifurcation Supercritical Hopf bifurcation Subcritical Hopf bifurcation Hyperchaos control 



This work is supported by Deanship of Scientific Research at Majmaah University. The author thanks the anonymous reviewers for providing some helpful comments which improve the style, readability and clarity of this work.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Basic Engineering Sciences, College of EngineeringMajmaah UniversityAl-MajmaahSaudi Arabia
  2. 2.Mansoura Higher Institute for Engineering and TechnologyMansouraEgypt

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