Abstract
This paper presents a new four-dimensional autonomous system having complex hyperchaotic dynamics. Basic properties of this new system are analyzed, and the complex dynamical behaviors are investigated by dynamical analysis approaches, such as time series, Lyapunov exponents’ spectra, bifurcation diagram, phase portraits. Moreover, when this new system is hyperchaotic, its two positive Lyapunov exponents are much larger than those of hyperchaotic systems reported before, which implies the new system has strong hyperchaotic dynamics in itself. The Kaplan–Yorke dimension, Poincaré sections and the frequency spectra are also utilized to demonstrate the complexity of the hyperchaotic attractor. It is also observed that the system undergoes an intermittent transition from period directly to hyperchaos. The statistical analysis of the intermittency transition process reveals that the mean lifetime of laminar state between bursts obeys the power-law distribution. It is shown that in such four-dimensional continuous system, the occurrence of intermittency may indicate a transition from period to hyperchaos not only to chaos, which provides a possible route to hyperchaos. Besides, the local bifurcation in this system is analyzed and then a Hopf bifurcation is proved to occur when the appropriate bifurcation parameter passes the critical value. All the conditions of Hopf bifurcation are derived by applying center manifold theorem and Poincaré–Andronov–Hopf bifurcation theorem. Numerical simulation results show consistency with our theoretical analysis.
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References
Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)
Clifford, M.J., Cox, S.M.: Smart baffle placement for chaotic mixing. Nonlinear. Dyn. 43, 117–126 (2006)
Lebaron, B., Arthur, W.B., Palmer, R.: Time series of properties of an artificial stock market. J. Econ. Dyn. Control 23, 1487–1516 (1999)
Ma, J.H., Cui, Y.Q., Liu, L.X.: A study on the complexity of a business cycle model with great excitements in non-resonant condition. Chaos Solitons Fractals 39, 2258–2267 (2009)
Parlitz, U., Kocarev, L., Stojanovski, T.: Encoding messages using chaotic synchronization. Phys. Rev. E 53, 4351–4361 (1996)
Chen, G.R., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9, 1465–1466 (1999)
Matsumoto, T.: A chaotic attractor from Chua’s circuit. IEEE Trans. Circuits Syst. I 31, 1055–1058 (1984)
Lü, J.H., Chen, G.R.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 12, 659–661 (2002)
Lü, J.H., Chen, G.R., Cheng, D.Z., Čelikovský, S.: Bridge the gap between the Lorenz system and the Chen system. Int. J. Bifurc. Chaos 12, 2917–2926 (2002)
Rössler, O.E.: An equation for hyperchaos. Phys. Lett. A 71, 155–157 (1979)
Cafagna, D., Grassi, G.: New 3D-scroll attractors in hyperchaotic Chua’s circuits forming a ring. Int. J. Bifurc. Chaos 13, 2889–2903 (2003)
Li, Y.X., Chen, G.R., Tang, W.K.S.: Controlling a unified chaotic system to hyperchaotic. IEEE Trans. Circuits Syst. II 52, 204–207 (2005)
Chen, Z.Q., Yang, Y., Qi, G.Y., Yuan, Z.Z.: A novel hyperchaos system only with one equilibrium. Phys. Lett. A 360, 696–701 (2007)
Gao, T.G., Chen, G.R., Chen, Z.Q., Cang, S.J.: The generation and circuit implementation of a new hyperchaos based upon Lorenz system. Phys. Lett. A 361, 78–86 (2007)
Qi, G.Y., Van Wyk, M.A., Van Wyk, B.J., Chen, G.R.: On a new hyperchaotic system. Phys. Lett. A 372, 124–136 (2008)
Qi, G.Y., Van Wyk, M.A., Van Wyk, B.J., Chen, G.R.: A new hyperchaotic system and its implementation. Chaos Solitons Fractals 40, 2544–2549 (2009)
Wu, W.J., Chen, Z.Q., Yuan, Z.Z.: The evolution of a novel four-dimensional autonomous system: among 3-torus, limit cycle, 2-torus, chaos and hyperchaos. Chaos Solitons Fractals 39, 2340–2356 (2009)
Wang, J.Z., Chen, Z.Q., Yuan, Z.Z.: The generation of a hyperchaotic system based on a three-dimensional autonomous chaotic system. Chin. Phys. 15, 1216–1225 (2006)
Qi, G.Y., Chen, G.R., Du, S.Z., Chen, Z.Q., Yuan, Z.Z.: Analysis of a new chaotic system. Phys. A 352, 295–308 (2005)
Sparrow, C.: The Lorenz Equations: Bifurcation, Chaos, and Strange Attractors. Springer, New York (1982)
Ueta, T., Chen, G.R.: Bifurcation analysis of Chen’s equation. Int. J. Bifurc. Chaos 10, 1917–1931 (2000)
Gao, Q., Ma, J.H.: Chaos and Hopf bifurcation of a finance system. Nonlinear Dyn. 58, 209–216 (2009)
Feigenbaum, M.J.: Quantitative universality for a class of non-linear transformations. J. Stat. Phys. 19, 25–52 (1978)
Pomeau, Y., Manneville, P.: Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74, 189–197 (1980)
Grebogi, C., Ott, E., Romeiras, F., Yorke, J.A.: Critical exponents for crisis-induced intermittency. Phys. Rev. A 36, 5365–5380 (1987)
Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971)
Yanchuk, S., Kapitaniak, T.: Chaos-hyperchaos transition in coupled Rössler systems. Phys. Lett. A 290, 139–144 (2001)
Zhou, Q., Chen, Z.Q., Yuan, Z.Z.: Blowout bifurcation and chaos–hyperchaos transition in five-dimensional continuous autonomous systems. Chaos Solitons Fractals 40, 1012–1020 (2009)
Wolf, A., Swift, J., Swinney, H., John, A.: Determining Lyapunov exponents from a time series. Physica D 16, 285–317 (1985)
Perez, G., Cerdeira, H.A.: Extracting messages masked by chaos. Phys. Rev. Lett. 74, 1970–1973 (1995)
Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1990)
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Wu, W., Chen, Z. Hopf bifurcation and intermittent transition to hyperchaos in a novel strong four-dimensional hyperchaotic system. Nonlinear Dyn 60, 615–630 (2010). https://doi.org/10.1007/s11071-009-9619-4
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DOI: https://doi.org/10.1007/s11071-009-9619-4