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Analytical and numerical study of Hopf bifurcation scenario for a three-dimensional chaotic system. (English) Zbl 1354.37041

Summary: 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.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37G10 Bifurcations of singular points in dynamical systems
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
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[1] Li, F., Jin, Y.L.: Hopf bifurcation analysis and numerical simulation in a 4-hyperchaotic system. Nonlinear Dyn. 67, 2857-2864 (2012) · Zbl 1251.34064 · doi:10.1007/s11071-011-0194-0
[2] Li, H.W., Wang, M.: Hopf bifurcation analysis in a Lorenz-type system. Nonlinear Dyn. 71, 235-240 (2012) · Zbl 1268.34076 · doi:10.1007/s11071-012-0655-0
[3] Yu, Y.G., Zhang, S.C.: Hopf bifurcation analysis of the Lu system. Chaos Solitons Fractals 21, 1215-1220 (2004) · Zbl 1061.37029 · doi:10.1016/j.chaos.2003.12.063
[4] Zhou, X.B., Wu, Y., Li, Y., Wei, Z.X.: Hopf bifurcation analysis of Liu system. Chaos Solitons Fractals 36, 1385-1391 (2008) · Zbl 1137.37321 · doi:10.1016/j.chaos.2006.09.008
[5] Sun, M., Tian, L.X., Yin, J.: Hopf bifurcation analysis of the energy resource chaotic system. Int. J. Nonlinear Sci. 1, 49-53 (2006) · Zbl 1394.37126
[6] Wang, X.: Si’lnikov chaos and Hopf bifurcation analysis of Rucklidge system. Chaos Solitons Fractals 42, 2208-2217 (2009) · Zbl 1198.37058 · doi:10.1016/j.chaos.2009.03.137
[7] Zhuang, K.J.: Hopf bifurcation analysis for a novel hyperchaotic system. J. Comput. Nonlinear Dyn. 8(1), 014501 (2012) · doi:10.1115/1.4006327
[8] Yan, Z.Y.: Hopf bifurcation in the Lorenz-type chaotic system. Chaos Solitons Fractals 31(5), 1135-1142 (2007) · Zbl 1140.37345 · doi:10.1016/j.chaos.2005.03.036
[9] Kuznetsov, Y.A.: Element of Applied Bifurcation Theory. Springer, New York (1998) · Zbl 0914.58025
[10] Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos: An Introduction to Dynamical System. Springer, New York (2000) · Zbl 0867.58043
[11] Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Application to Physics, Biology, Chemistry, and Engineering. Westview, Cambridge (2000)
[12] Marsden, J.E., McCracken, M.: The Hopf Bifurcation and Its Application. Springer, New York (1976) · Zbl 0346.58007 · doi:10.1007/978-1-4612-6374-6
[13] Guckenheimer, J., Myers, M., Sturmfels, B.: Computing Hopf bifurcation I. SIAM J. Numer. Anal. 34(1), 1-21 (2006) · Zbl 0948.37037 · doi:10.1137/S0036142993253461
[14] Garrat, T.J., Moore, G., Spence, A.: Bifurcation & Chaos: Analysis, Algorithms, Application. Birkhäuser Basel, Switzerland (1991)
[15] Gao, Q., Ma, J.H.: Chaos and Hopf bifurcation of a finance system. Nonlinear Dyn. 58, 209-216 (2009) · Zbl 1183.91193 · doi:10.1007/s11071-009-9472-5
[16] Aqeel, M., Yue, B.Z.: Nonlinear analysis of stretch-twist-fold (STF) flow. Nonlinear Dyn. 72, 581-590 (2013) · doi:10.1007/s11071-012-0736-0
[17] Itik, M., Banks, S.P.: Chaos in a three-dimensional cancer model. Int. J. Bifurc. Chaos 20, 71-79 (2010) · Zbl 1183.34064 · doi:10.1142/S0218127410025417
[18] Zhao, Z., Yang, L., Chen, L.: Bifurcation & chaos of biochemical reaction model with impulsive perturbation. Nonlinear Dyn. 63, 521-535 (2010) · doi:10.1007/s11071-010-9722-6
[19] 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) · Zbl 1075.94527 · doi:10.1016/j.chaos.2004.09.077
[20] Mittal, A.K., Mukerjee, S., Shukla, R.P.: Bifurcation analysis of some forced Lu systems. Commun. Nonlinear Sci. Numer. Simul. 16, 789-797 (2011) · Zbl 1221.37066
[21] Ueta, T., Chen, G.R.: Bifurcation analysis of Chen’s equation. Int. J. Bifurc. Chaos 10, 1917-1931 (2000) · Zbl 1090.37531
[22] Yue, B.Z., Aqeel, M.: Chaotification in the stretch-twist-fold (STF) flow. Chin. Sci. Bull. 58, 1655-1662 (2013) · doi:10.1007/s11434-013-5754-x
[23] 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 · Zbl 1238.34095
[24] Bhalekar, S.: Forming mechanism of Bhalekar-Gejji chaotic dynamical system. Am. J. Comput. Appl. Math. 2(6), 257-259 (2012) · Zbl 1137.37321
[25] Ermentrout, B.: Simulating, Analyzing, and Animating Dynamical Systems. SIAM, Pennsylvania (2002) · Zbl 1003.68738 · doi:10.1137/1.9780898718195
[26] Parker, T.S., Chua, L.O.: Practical Numerical Algorithm for Chaotic System. Springer, Berlin (1989) · Zbl 0692.58001 · doi:10.1007/978-1-4612-3486-9
[27] Seydel, R.: Practical Bifurcation and Stability Analysis, from Equilibrium to Chaos. Springer, New York (1994) · Zbl 0806.34028
[28] 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) · Zbl 0876.65032 · doi:10.1142/S0218127491000397
[29] Beyn, WJ; Champneys, A.; Doedel, EJ; Kuznetov, YA; Govaerts, W.; Sandstede, B.; Fiedler, B. (ed.), Numerical continuation and computation of normal forms, No. 2, 149-219 (2001), Amsterdam
[30] Friedman, M.J., Doel, E.J.: Numerical computation and continuation of invariant manifolds connecting fixed points. SIAM J. Numer. Anal. 28, 789-808 (1991) · Zbl 0735.65054 · doi:10.1137/0728042
[31] Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, New York (1981) · Zbl 0474.34002
[32] Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1990) · Zbl 0701.58001 · doi:10.1007/978-1-4757-4067-7
[33] Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. Wiley, New York (1995) · Zbl 0848.34001 · doi:10.1002/9783527617548
[34] Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York (1983) · Zbl 0515.34001 · doi:10.1007/978-1-4612-1140-2
[35] Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory. Applied Mathematical Sciences series, vol. 51. Springer, Berlin (1984) · Zbl 0607.35004
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