Abstract
In this article, Hopf bifurcation is characterized for newly proposed Bhalekar–Gejji three-dimensional chaotic dynamical system. By analytical method, a sufficient condition is established for the existence of Hopf bifurcation. Using numerical continuation technique, Hopf bifurcation diagram is analyzed for chaotic parameter which strengthens our analytical results. Moreover, influence of system parameters on dynamical behavior is investigated using phase portraits, Lyapunov exponents, Lyapunov dimensions and Poincaré maps. Theoretical analysis and numerical simulations demonstrate the rich dynamics of the system.
Similar content being viewed by others
References
Li, F., Jin, Y.L.: Hopf bifurcation analysis and numerical simulation in a 4-hyperchaotic system. Nonlinear Dyn. 67, 2857–2864 (2012)
Li, H.W., Wang, M.: Hopf bifurcation analysis in a Lorenz-type system. Nonlinear Dyn. 71, 235–240 (2012)
Yu, Y.G., Zhang, S.C.: Hopf bifurcation analysis of the Lu system. Chaos Solitons Fractals 21, 1215–1220 (2004)
Zhou, X.B., Wu, Y., Li, Y., Wei, Z.X.: Hopf bifurcation analysis of Liu system. Chaos Solitons Fractals 36, 1385–1391 (2008)
Sun, M., Tian, L.X., Yin, J.: Hopf bifurcation analysis of the energy resource chaotic system. Int. J. Nonlinear Sci. 1, 49–53 (2006)
Wang, X.: Si’lnikov chaos and Hopf bifurcation analysis of Rucklidge system. Chaos Solitons Fractals 42, 2208–2217 (2009)
Zhuang, K.J.: Hopf bifurcation analysis for a novel hyperchaotic system. J. Comput. Nonlinear Dyn. 8(1), 014501 (2012)
Yan, Z.Y.: Hopf bifurcation in the Lorenz-type chaotic system. Chaos Solitons Fractals 31(5), 1135–1142 (2007)
Kuznetsov, Y.A.: Element of Applied Bifurcation Theory. Springer, New York (1998)
Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos: An Introduction to Dynamical System. Springer, New York (2000)
Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Application to Physics, Biology, Chemistry, and Engineering. Westview, Cambridge (2000)
Marsden, J.E., McCracken, M.: The Hopf Bifurcation and Its Application. Springer, New York (1976)
Guckenheimer, J., Myers, M., Sturmfels, B.: Computing Hopf bifurcation I. SIAM J. Numer. Anal. 34(1), 1–21 (2006)
Garrat, T.J., Moore, G., Spence, A.: Bifurcation & Chaos: Analysis, Algorithms, Application. Birkhäuser Basel, Switzerland (1991)
Gao, Q., Ma, J.H.: Chaos and Hopf bifurcation of a finance system. Nonlinear Dyn. 58, 209–216 (2009)
Aqeel, M., Yue, B.Z.: Nonlinear analysis of stretch-twist-fold (STF) flow. Nonlinear Dyn. 72, 581–590 (2013)
Itik, M., Banks, S.P.: Chaos in a three-dimensional cancer model. Int. J. Bifurc. Chaos 20, 71–79 (2010)
Zhao, Z., Yang, L., Chen, L.: Bifurcation & chaos of biochemical reaction model with impulsive perturbation. Nonlinear Dyn. 63, 521–535 (2010)
Li, S.J., Alvarez, G., Chen, G.R.: Breaking a chaos-based secure communication scheme designed by an improved modulation method. Chaos Solitons Fractals 25, 109–120 (2005)
Mittal, A.K., Mukerjee, S., Shukla, R.P.: Bifurcation analysis of some forced Lu systems. Commun. Nonlinear Sci. Numer. Simul. 16, 789–797 (2011)
Ueta, T., Chen, G.R.: Bifurcation analysis of Chen’s equation. Int. J. Bifurc. Chaos 10, 1917–1931 (2000)
Yue, B.Z., Aqeel, M.: Chaotification in the stretch-twist-fold (STF) flow. Chin. Sci. Bull. 58, 1655–1662 (2013)
Bhalekar, S., Daftardar-Gejji, V.: A new chaotic dynamical system and its synchronization. In: Proceedings of the International Conference on Mathematical Sciences in Honor of Prof. A. M. Mathai, 3–5 January 2011, Palai, Kerla-686 574, India
Bhalekar, S.: Forming mechanism of Bhalekar-Gejji chaotic dynamical system. Am. J. Comput. Appl. Math. 2(6), 257–259 (2012)
Ermentrout, B.: Simulating, Analyzing, and Animating Dynamical Systems. SIAM, Pennsylvania (2002)
Parker, T.S., Chua, L.O.: Practical Numerical Algorithm for Chaotic System. Springer, Berlin (1989)
Seydel, R.: Practical Bifurcation and Stability Analysis, from Equilibrium to Chaos. Springer, New York (1994)
Doedel, E.J., Keller, H.B., Kernevez, J.P.: Numerical analysis and control of bifurcation problems (I), bifurcation in finite dimension. Int. J. Bifurc. Chaos 1, 493–520 (1991)
Beyn, W.J., Champneys, A., Doedel, E.J., Kuznetov, Y.A., Govaerts, W., Sandstede, B.: Numerical continuation and computation of normal forms. In: Fiedler, B. (ed.) Handbook of Dynamical Systems, vol. 2, pp. 149–219. Elsevier, Amsterdam (2001)
Friedman, M.J., Doel, E.J.: Numerical computation and continuation of invariant manifolds connecting fixed points. SIAM J. Numer. Anal. 28, 789–808 (1991)
Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, New York (1981)
Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1990)
Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. Wiley, New York (1995)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York (1983)
Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory. Applied Mathematical Sciences series, vol. 51. Springer, Berlin (1984)
Acknowledgments
The authors express their sincere gratitude to the reviewers for their valuable suggestions and comments to improve the quality of the manuscript. This work is partially supported by Higher Education Commission (HEC), Pakistan under the program “Startup Research Grant” with Grant No. SRGP (PD-IPFP/HRD/ HEC/2014/1659.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Aqeel, M., Ahmad, S. Analytical and numerical study of Hopf bifurcation scenario for a three-dimensional chaotic system. Nonlinear Dyn 84, 755–765 (2016). https://doi.org/10.1007/s11071-015-2525-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-015-2525-z