Abstract
The generation of n-scroll chaotic attractors by using saturated nonlinear function series (SNFS) realized with positive-type second generation current conveyors (CCII+s), is introduced. The nonlinear dynamical system is expressed by a third-order differential equation and to carry out numerical simulations, SNFS are ideally modeled by using staircase functions. Therefore, numerical simulations are introduced to approximate the swings, widths, breakpoints and equilibrium points of the n-scroll attractors by considering, as input variables: the dynamic range associated to active devices, gain of the nonlinear system and the number of scrolls. Therefore, its dynamical behavior is investigated in the state space. Besides, the CCII± is a versatile analog building block and it has been demonstrated to be very useful in several linear and nonlinear applications, since CCII-based implementations offer better performances that Opamps-based implementations in terms of accuracy and bandwidth. Therefore, the nonlinear system is synthesized with CCII+s to generate 3- and 4-scrolls. HSPICE simulations and experimental results are shown to verify the agreement on the behavior of the proposed circuit and the numerical simulations.
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References
AD844 Data sheet, wwww.analogdevices.org (2003)
Arena, P., Baglio, S., Fortuna, L., Manganaro, G.: Generation of n-double scrolls via cellular neural networks. Int. J. Bifurc. Chaos 24(3), 5 (1996)
Aziz-Alaoui, M.A., Robert, C., Grebogi, C.: Dynamics of a Henon–Lozi-type map. Chaos Solitons Fractals 12(12), 18 (2001)
Cafagna, D., Grassi, G.: Decomposition method for studying smooth Chua’s equation with application to hyperchaotic multiscroll attractors. Int. J. Bifurc. Chaos 17(11), 18 (2007)
Chua, L.O., Komuro, M., Matsumoto, T.: The double scroll family. IEEE Trans. Circuits Syst. 33(11), 48 (1986)
Elwakil, A.S., Kennedy, M.P.: Three-phase oscillator modified for chaos. Microelectron. J. 30(9), 5 (1999)
Elwakil, A.S., Kennedy, M.P.: Improved implementation of Chua’s chaotic oscillator using current feedback Opamp. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47(1), 4 (2000)
Elwakil, A.S., Kennedy, M.P.: Chua’s circuit decomposition: a systematic design approach for chaotic oscillators. J. Franklin Inst. 337, 6 (2000)
Elwakil, A.S., Kennedy, M.P.: Construction of classes of circuit independent chaotic oscillators using passive-only nonlinear devices. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 48(3), 19 (2001)
Elwakil, A.S., Soliman, A.M.: Current conveyor chaos generators. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 46(3), 6 (1999)
Elwakil, A.S., Salama, K.N., Kennedy, M.P.: An equation for generating chaos and its monolithic implementation. Int. J. Bifurc. Chaos 12(12), 11 (2002)
Gámez-Guzmán, L., Cruz-Hernández, C., López-Gutiérrez, R.M., García-Guerrero, E.: Synchronization of Chua’s circuits with multi-scroll attractors: application to communication. Commun. Nonlinear Sci. Numer. Simul. 14(6), 11 (2009). doi:10.1016/j.cnsns.2008.10.009
Gandhi, G.: An improved Chua’s circuit and its use in hyperchaotic circuit. Analog Integr. Circuits Signal Process. 46(2), 6 (2006)
Han, F., Yu, X., Wang, Y., Feng, Y., Chen, G.: N-scroll chaotic oscillators by second-order systems and double hysteresis blocks. Electron. Lett. 39(23), 3 (2003)
Hasan, S., Khan, I.A.: Multi-scroll with current conveyor. In: IEEE Multimedia, Signal Processing and Communication Technologies, vol. 4 (2009)
Liu, S.I., Wu, D.S., Tsao, H.W., Wu, J., Tsay, J.H.: Nonlinear circuit applications with current conveyors. IEEE Proc. G 140(1), 6 (1993)
Lü, J., Chen, G.: Generating multiscroll chaotic attractors: Theories, methods and applications. Int. J. Bifurc. Chaos 16(4), 84 (2006)
Lü, J., Chen, G., Yu, X., Leung, H.: Generating multi-scroll chaotic attractors via switching control. In: Proc. 5th Asian Control Conf., p. 9, July 20–23, Melbourne, Australia (2004)
Lü, J., Chen, G., Yu, X., Leung, H.: Design and analysis of multi-scroll chaotic attractors from saturated function series. IEEE Trans. Circuits Syst. I 51(12), 15 (2004)
Lü, J., Yu, S., Leung, H., Chen, G.: Experimental verification of multidirectional multiscroll chaotic attractors. IEEE Trans. CAS-I 53(1), 17 (2006)
Muñoz-Pacheco, J.M., Tlelo-Cuautle, E.: Synthesis of n-scroll attractors using saturated functions from high-level simulation. J. Phys. Conf. Ser. 96(1), 10 (2008)
Muñoz-Pacheco, J.M., Tlelo-Cuautle, E.: Automatic synthesis of 2D-n-scrolls chaotic systems by behavioral modeling. J. Appl. Res. Technol. 7(1), 10 (2009)
Ozoguz, S., Elwakil, A.S., Salama, K.N.: N-scroll chaos generator using nonlinear transconductor. Electron. Lett. 38(3), 2 (2002)
Radman, A.G., Soliman, A.M., El-Sedeek, A.L.: MOS realization of the double-scroll-like chaotic equation. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 50(2), 4 (2003)
Sánchez-López, C., Castro-Hernández, A., Pérez-Trejo, A.: Experimental verification of the Chua’s circuit designed with UGCs. IEICE Electron. Express 5(17), 5 (2008)
Sánchez-López, C., Tlelo-Cuautle, E., Carrasco-Aguilar, M.A., Morales-López, F.E., Cante-Michcol, B.: Multi-scroll chaotic oscillator employing UGCs. In: IEEE CONIELECOMP, Mexico, vol. 4 (2009)
Smith, K.C., Sedra, A.: The current conveyor—a new circuit building block. IEEE Proc. 56(8), 2 (1968)
Smith, K.C., Sedra, A.: Realization of the Chua family of new nonlinear network elements using the current conveyor. IEEE Trans. Circuit Theory 17(1), 3 (1970)
Sprott, J.C.: Simple chaotic systems and circuits. Am. J. Phys. 68, 6 (2000)
Stavroulakis, P.: Chaos Applications in Telecommunications. CRC Press, Boca Raton (2005)
Suykens, J.A.K., Vandewalle, J.: Generation of n-double scrolls (n=1,2,3,4,…). IEEE Trans. CAS-I 40(11), 8 (1993)
Tang, K.S., Zhong, G.Q., Chen, G., Man, K.F.: Generation of n-scroll attractors via sine function. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 48(11), 4 (2001)
Tlelo-Cuautle, E., Muñoz-Pacheco, J.M., Martínez Carballido, J.: Frequency-scaling simulation of Chua’s circuit by automatic determination and control of step-size. Appl. Math. Comput. 194(2), 6 (2007)
Trejo-Guerra, R., Tlelo-Cuautle, E., Cruz Hernández, C., Sánchez-López, C., Fakhfakh, M.: Current conveyor realization of synchronized Chua’s circuits for binary communications. IEEE DTIS, vol. 4 (2008)
Trejo-Guerra, R., Tlelo-Cuautle, E., Cruz-Hernández, C., Sánchez-López, C.: Chaotic communication system using Chua’s oscillators realized with CCII+s, Int. J. Bifurc. Chaos 19(12) (2009, to be published)
Wang, F.Q., Liu, C.X.: Generation of multi-scroll chaotic attractors via the saw-tooth function. Int. J. Mod. Phys. 22(15), 7 (2008)
Yalcin, M.E., Suykens, J.A.K., Vandewalle, J.: Experimental confirmation of 3- and 5-scroll attractors from a generalized Chua’s circuit. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47(3), 5 (2000)
Yalcin, M.E., Suykens, J.A.K., Vandewalle, J., Ozoguz, S.: Families of scroll grid attractors. Int. J. Bifurc. Chaos 12(1), 19 (2002)
Yu, S.M., Lü, J., Leung, H., Chen, G.: Design and implementation of n-scroll chaotic attractors from a general jerk circuit. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 52(7), 19 (2005)
Yu, S., Tang, W.K.S., Chen, G.: Generation of n×m-scroll attractors under a Chua-circuit framework. Int. J. Bifurc. Chaos 17(11), 14 (2007)
Zhang, H., Li, C., Zhang, J., Liao, X., Yu, J.: Controlling chaotic Chua’s circuit based on piecewise quadratic Lyapunov functions method. Chaos Solitons Fractals 22(5), 9 (2004)
Zhong, G., Man, K.F., Chen, G.: A systematic approach to generating n-scroll attractors. Int. J. Bifurc. Chaos 12(12), 9 (2002)
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Sánchez-López, C., Trejo-Guerra, R., Muñoz-Pacheco, J.M. et al. N-scroll chaotic attractors from saturated function series employing CCII+s. Nonlinear Dyn 61, 331–341 (2010). https://doi.org/10.1007/s11071-009-9652-3
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DOI: https://doi.org/10.1007/s11071-009-9652-3