Fractional Order of a New 7D Hyperchaotic Lorenz-like System

Year 2021, Volume: 9 Issue: 1, 76 - 89, 28.04.2021

Emrah Haspolat , Bengi Yıldız

Abstract

In this paper, a new 7D hyperchaotic Lorenz-like system is proposed with perspective of fractional order. Numerical implementations of this proposed system with specific parameters are investigated and compared with the new 7D continuous hyperchaotic system. In addition to this, due to the hyperchaotic attractors do not exist lower than 0.6, the values of fractional order are analysed in range between 0.6 to 1. Stability conditions are obtained through the stability theory of fractional systems. Numerical analysis of Lyapunov exponents verifies the existence of hyperchaos for less than five orders.

Keywords

Chaos, Fractional order derivatives, Fractional stability, Hyperchaos, Lorenz-like system, Lyapunov exponents

References

• [1] Arena, P., Baglio, S., Fortuna, L., and Manganaro, G. (1995). Hyperchaos from cellular neural networks. Electronics letters, 31(4):250–251.
• [2] Benettin, G., Galgani, L., Giorgilli, A., and Strelcyn, J.-M. (1980). Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. part 1: Theory. Meccanica, 15(1):9–20.
• [3] Cannas, B. and Cincotti, S. (2002). Hyperchaotic behaviour of two bi-directionally chua’s circuits. International Journal of Circuit Theory and Applications, 30:625 – 637.
• [4] Caputo, M. (1967). Linear models of dissipation whose q is almost frequency independent—ii. Geophysical Journal International, 13(5):529–539.
• [5] Carpinteri, A. and Mainardi, F. (2014). Fractals and fractional calculus in continuum mechanics, volume 378. Springer.
• [6] Chen, Z., Yang, Y., Qi, G., and Yuan, Z. (2007). A novel hyperchaos system only with one equilibrium. Physics Letters A, 360(6):696–701.
• [7] Debnath, L. (2003). Recent applications of fractional calculus to science and engineering. International Journal of Mathematics and Mathematical Sciences, 2003(54):3413–3442.
• [8] Diethelm, K. and Ford, N. J. (2002). Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications, 265(2):229–248.
• [9] Diethelm, K., Ford, N. J., and Freed, A. D. (2002). A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics, 29(1-4):3–22.
• [10] Duan, J.-S. (2005). Time-and space-fractional partial differential equations. Journal of mathematical physics, 46(1):013504.
• [11] Farghaly, A. and Shoreh, A. (2018). Some complex dynamical behaviors of the new 6d fractional-order hyperchaotic lorenz-like system. Journal of the Egyptian Mathematical Society, 26(1):138–155.
• [12] Garrappa, R. (2011). Predictor-corrector pece method for fractional differential equations. MATLAB Central File Exchange.
• [13] Hartley, T. T., Lorenzo, C. F., and Qammer, H. K. (1995). Chaos in a fractional order chua’s system. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 42(8):485–490.
• [14] Hilfer, R. et al. (2000). Applications of fractional calculus in physics, volume 35. World scientific Singapore.
• [15] Hsieh, J.-Y., Hwang, C.-C., Wang, A.-P., and Li, W.-J. (1999). Controlling hyperchaos of the rossler system. International Journal of Control, 72(10):882–886.
• [16] Huang, X., Zhao, Z., Wang, Z., and Li, Y. (2012). Chaos and hyperchaos in fractional-order cellular neural networks. Neurocomputing, 94:13–21.
• [17] Jajarmi, A., Hajipour, M., Mohammadzadeh, E., and Baleanu, D. (2018). A new approach for the nonlinear fractional optimal control problems with external persistent disturbances. Journal of the Franklin Institute, 355(9):3938–3967.
• [18] Kapitaniak, T. and Chua, L. O. (1994). Hyperchaotic attractors of unidirectionally-coupled chua’s circuits. International Journal of Bifurcation and Chaos, 4(02):477–482.
• [19] Kilbas, A. A., Srivastava, H. M., and Trujillo, J. J. (2006). Theory and applications of fractional differential equations, volume 204. elsevier.
• [20] Li, C. and Chen, G. (2004). Chaos in the fractional order chen system and its control. Chaos, Solitons & Fractals, 22(3):549–554.
• [21] Li, Y., Chen, Y., and Podlubny, I. (2010). Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized mittag–leffler stability. Computers & Mathematics with Applications, 59(5):1810–1821.
• [22] Lu, J. G. (2006). Chaotic dynamics of the fractional-order lu system and its synchronization. Physics Letters A, 354(4):305–311.
• [23] Lu, J. G. and Chen, G. (2006). A note on the fractional-order chen system. Chaos, Solitons & Fractals, 27(3):685–688.
• [24] Luo, C. and Wang, X. (2013). Chaos in the fractional-order complex lorenz system and its synchronization. Nonlinear Dynamics, 71(1-2):241–257.
• [25] Magin, R. L. (2006). Fractional calculus in bioengineering, volume 2. Begell House Redding.
• [26] Matignon, D. (1996). Stability results for fractional differential equations with applications to control processing. In Computational engineering in systems applications, volume 2, pages 963–968. Lille, France.
• [27] Matouk, A. (2009). Stability conditions, hyperchaos and control in a novel fractional order hyperchaotic system. Physics Letters A, 373(25):2166–2173.
• [28] Metzler, R. and Klafter, J. (2000). The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics reports, 339(1):1–77.
• [29] Miller, K. S. and Ross, B. (1993). An introduction to the fractional calculus and fractional differential equations. Wiley.
• [30] Monje, C. A., Chen, Y., Vinagre, B. M., Xue, D., and Feliu-Batlle, V. (2010). Fractional-order systems and controls: fundamentals and applications. Springer Science & Business Media.
• [31] Oldham, K. and Spanier, J. (1974). The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier.
• [32] Podlubny, I. (1998). Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier.
• [33] Ross, B. (2006). Fractional calculus and its applications: proceedings of the international conference held at the University of New Haven, June 1974, volume 457. Springer.
• [34] Rossler, O. E. (1979). An equation for hyperchaos. Physics Letters A, 71(2-3):155–157.
• [35] Udaltsov, V., Goedgebuer, J., Larger, L., Cuenot, J., Levy, P., and Rhodes, W. (2003). Communicating with hyperchaos: the dynamics of a dnlf emitter and recovery of transmitted information. Optics and Spectroscopy, 95(1):114–118.
• [36] Ullah, S., Khan, M. A., Farooq, M., Gul, T., and Hussain, F. (2020). A fractional order hbv model with hospitalization. Discrete & Continuous Dynamical Systems-S, 13(3):957.
• [37] Vicente, R., Dauden, J., Colet, P., and Toral, R. (2005). Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop. IEEE Journal of Quantum Electronics, 41(4):541–548.
• [38] Wang, S. and Wu, R. (2017). Dynamic analysis of a 5d fractional-order hyperchaotic system. International Journal of Control, Automation and Systems, 15(3):1003–1010.
• [39] Wolf, A., Swift, J. B., Swinney, H. L., and Vastano, J. A. (1985). Determining lyapunov exponents from a time series. Physica D: Nonlinear Phenomena, 16(3):285–317.
• [40] Yang, Q. and Bai, M. (2017). A new 5d hyperchaotic system based on modified generalized lorenz system. Nonlinear Dynamics, 88(1):189–221.
• [41] Yang, Q. and Chen, C. (2013). A 5d hyperchaotic system with three positive lyapunov exponents coined. International Journal of Bifurcation and Chaos, 23(06):1350109.
• [42] Yang, Q., Osman, W. M., and Chen, C. (2015). A new 6d hyperchaotic system with four positive lyapunov exponents coined. International Journal of Bifurcation and Chaos, 25(04):1550060.
• [43] Yang, Q., Zhang, K., and Chen, G. (2009). Hyperchaotic attractors from a linearly controlled lorenz system. Nonlinear Analysis: Real World Applications, 10(3):1601–1617.
• [44] Yang, Q., Zhu, D., and Yang, L. (2018). A new 7d hyperchaotic system with five positive lyapunov exponents coined. International Journal of Bifurcation and Chaos, 28(05):1850057.
• [45] Zhang, W., Zhou, S., Li, H., and Zhu, H. (2009). Chaos in a fractional-order rossler¨ system. Chaos, Solitons & Fractals, 42(3):1684–1691.