Modified jerk system with self-exciting and hidden flows and the effect of time delays on existence of multi-stability

  • Karthikeyan Rajagopal
  • Sajad Jafari
  • Akif Akgul
  • Anitha Karthikeyan
Original Paper
  • 57 Downloads

Abstract

In this paper, we report a new chaotic jerk system which shows self-excited and hidden oscillations depending on its parameters. Dynamic analysis shows that the proposed system exhibits multi-stability and coexisting attractors. To study the effect of time delays on the multi-stability feature of the system, we introduce multiple time delays in the third state variable. Investigation of dynamical properties of the time-delayed system shows the disappearance of multi-stability. Such a feature has not been reported earlier in the literatures.

Keywords

Jerk systems Hidden oscillations Bifurcation Multi-stability Time-delayed systems FPGA implementation 

References

  1. 1.
    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)CrossRefMATHGoogle Scholar
  2. 2.
    Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57(5), 397–398 (1976)CrossRefMATHGoogle Scholar
  3. 3.
    Rössler, O.E.: Continuous chaos—four prototype equations. Ann. N. Y. Acad. Sci. 316(1), 376–392 (1979)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Sprott, J.C.: Some simple chaotic flows. Phys. Rev. E 50(2), R647 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Schot, S.H.: Jerk: the time rate of change of acceleration. Am. J. Phys. 46(11), 1090–1094 (1978)CrossRefGoogle Scholar
  6. 6.
    Kengne, J., Njitacke, Z.T., Fotsin, H.B.: Dynamical analysis of a simple autonomous jerk system with multiple attractors. Nonlinear Dyn. 83(1–2), 751–765 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kengne, J., Njikam, S.M., Signing, V.R.F.: A plethora of coexisting strange attractors in a simple jerk system with hyperbolic tangent nonlinearity. Chaos Solitons Fractals 106, 201–213 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kengne, J., Negou, A.N., Njitacke, Z.T.: Antimonotonicity, chaos and multiple attractors in a novel autonomous jerk circuit. Int. J. Bifurc. Chaos 27(07), 1750100 (2017)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Sprott, J.C.: Some simple chaotic jerk functions. Am. J. Phys. 65(6), 537–543 (1997)CrossRefGoogle Scholar
  10. 10.
    Sprott, J.C.: Simplest dissipative chaotic flow. Phys. Lett. A 228(4–5), 271–274 (1997)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Munmuangsaen, B., Srisuchinwong, B.: Elementary chaotic snap flows. Chaos Solitons Fractals 44(11), 995–1003 (2011)CrossRefGoogle Scholar
  12. 12.
    Munmuangsaen, B., Srisuchinwong, B., Sprott, J.C.: Generalization of the simplest autonomous chaotic system. Phys. Lett. A 375(12), 1445–1450 (2011)CrossRefMATHGoogle Scholar
  13. 13.
    Sprott, J.C.: A new chaotic jerk circuit. IEEE Trans. Circuits Syst. II Express Briefs 58(4), 240–243 (2011)CrossRefGoogle Scholar
  14. 14.
    Vaidyanathan, S., Volos, C., Pham, V.-T., Madhavan, K.: Analysis, adaptive control and synchronization of a novel 4-D hyperchaotic hyperjerk system and its spice implementation. Arch. Control Sci. 25(1), 135–158 (2015)MathSciNetGoogle Scholar
  15. 15.
    Vaidyanathan, S., Akgul, A., Kaçar, S., Çavuşoğlu, U.: A new 4-D chaotic hyperjerk system, its synchronization, circuit design and applications in RNG, image encryption and chaos-based steganography. Eur. Phys. J. Plus 133(2), 46 (2018)CrossRefGoogle Scholar
  16. 16.
    Ren, S., Panahi, S., Rajagopal, K., Akgul, A., Pham, V.-T., Jafari, S.: A new chaotic flow with hidden attractor: the first hyperjerk system with no equilibrium. Z. Naturforsch. A 73(3), 239–249 (2018)CrossRefGoogle Scholar
  17. 17.
    Yu, S., Lu, J., Leung, H., Chen, G.: Design and implementation of n-scroll chaotic attractors from a general jerk circuit. IEEE Trans. Circuits Syst. I Regul. Pap. 52(7), 1459–1476 (2005)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Yalçin, M.E.: Multi-scroll and hypercube attractors from a general jerk circuit using Josephson junctions. Chaos Solitons Fractals 34(5), 1659–1666 (2007)CrossRefGoogle Scholar
  19. 19.
    Chunxia, L., Jie, Y., Xiangchun, X., Limin, A., Yan, Q., Yongqing, F.: Research on the multi-scroll chaos generation based on jerk mode. Procedia Eng. 29, 957–961 (2012)CrossRefGoogle Scholar
  20. 20.
    Srisuchinwong, B., Nopchinda, D.: Current-tunable chaotic jerk oscillator. Electron. Lett. 49(9), 587–589 (2013)CrossRefGoogle Scholar
  21. 21.
    Volos, C., Akgul, A., Pham, V.T., Stouboulos, I., Kyprianidis, I.: A simple chaotic circuit with a hyperbolic sine function and its use in a sound encryption scheme. Nonlinear Dyn. 89(2), 1047–1061 (2017)CrossRefGoogle Scholar
  22. 22.
    Leonov, G.A., Kuznetsov, N.V., Vagaitsev, V.I.: Localization of hidden chuas attractors. Phys. Lett. A 375(23), 2230–2233 (2011)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Leonov, G.A., Kuznetsov, N.V., Vagaitsev, V.I.: Hidden attractor in smooth chua systems. Physica D 241(18), 1482–1486 (2012)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Leonov, G.A., Kuznetsov, N.V.: Hidden attractors in dynamical systems. from hidden oscillations in Hilbert–Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in chua circuits. Int. J. Bifurc Chaos 23(01), 1330002 (2013)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Leonov, G.A., Kuznetsov, N.V., Mokaev, T.N.: Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity. Commun. Nonlinear Sci. Numer. Simul. 28(1), 166–174 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Leonov, G.A., Kuznetsov, N.V., Mokaev, T.N.: Homoclinic orbits, and self-excited and hidden attractors in a lorenz-like system describing convective fluid motion. Eur. Phys. J. Spec. Topics 224(8), 1421–1458 (2015)CrossRefGoogle Scholar
  27. 27.
    Sharma, P.R., Shrimali, M.D., Prasad, A., Kuznetsov, N.V., Leonov, G.A.: Control of multistability in hidden attractors. Eur. Phys. J. Spec. Topics 224(8), 1485–1491 (2015)CrossRefGoogle Scholar
  28. 28.
    Sharma, P.R., Shrimali, M.D., Prasad, A., Kuznetsov, N.V., Leonov, G.A.: Controlling dynamics of hidden attractors. Int. J. Bifurc. Chaos 25(04), 1550061 (2015)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N.V., Leonov, G.A., Prasad, A.: Hidden attractors in dynamical systems. Phys. Rep. 637, 1–50 (2016)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Jafari, S., Sprott, J.C., Golpayegani, S.M.R.H.: Elementary quadratic chaotic flows with no equilibria. Phys. Lett. A 377(9), 699–702 (2013)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Jafari, S., Sprott, J.C., Pham, V.-T., Golpayegani, S.M.R.H., Jafari, A.H.: A new cost function for parameter estimation of chaotic systems using return maps as fingerprints. Int. J. Bifurc. Chaos 24(10), 1450134 (2014)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Pham, V.-T., Volos, C., Jafari, S., Wei, Z., Wang, X.: Constructing a novel no-equilibrium chaotic system. Int. J. Bifurc. Chaos 24(05), 1450073 (2014)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Tahir, F.R., Jafari, S., Pham, V.-T., Volos, C., Wang, X.: A novel no-equilibrium chaotic system with multiwing butterfly attractors. Int. J. Bifurc. Chaos 25(04), 1550056 (2015)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Jafari, S., Pham, V.-T., Kapitaniak, T.: Multiscroll chaotic sea obtained from a simple 3D system without equilibrium. Int. J. Bifurc. Chaos 26(02), 1650031 (2016)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Lao, S.-K., Shekofteh, Y., Jafari, S., Sprott, J.C.: Cost function based on gaussian mixture model for parameter estimation of a chaotic circuit with a hidden attractor. Int. J. Bifurc. Chaos 24(01), 1450010 (2014)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Pham, V.-T., Vaidyanathan, S., Volos, C., Jafari, S., Kingni, S.T.: A no-equilibrium hyperchaotic system with a cubic nonlinear term. Optik Int. J. Light Electron Opt. 127(6), 3259–3265 (2016)CrossRefGoogle Scholar
  37. 37.
    Wei, Z., Moroz, I., Sprott, J.C., Akgul, A., Zhang, W.: Hidden hyperchaos and electronic circuit application in a 5d self-exciting homopolar disc dynamo. Chaos Interdiscip. J. Nonlinear Sci. 27(3), 033101 (2017)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Wang, Z., Akgul, A., Pham, V.T., Jafari, S.: Chaos-based application of a novel no-equilibrium chaotic system with coexisting attractors. Nonlinear Dyn. 89(3), 1877–1887 (2017)CrossRefGoogle Scholar
  39. 39.
    Pham, V.-T., Vaidyanathan, S., Volos, C.K., Jafari, S., Kuznetsov, N.V., Hoang, T.M.: A novel memristive time-delay chaotic system without equilibrium points. Eur. Phys. J. Spec. Topics 225(1), 127–136 (2016)CrossRefGoogle Scholar
  40. 40.
    Molaie, M., Jafari, S., Sprott, J.C.: Simple chaotic flows with one stable equilibrium. Int. J. Bifurc. Chaos 23(11), 1350188 (2013)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Kingni, S.T., Jafari, S., Simo, H., Woafo, P.: Three-dimensional chaotic autonomous system with only one stable equilibrium: analysis, circuit design, parameter estimation, control, synchronization and its fractional-order form. Eur. Phys. J. Plus 129(5), 1–16 (2014)CrossRefGoogle Scholar
  42. 42.
    Pham, V.-T., Jafari, S., Volos, C., Giakoumis, A., Vaidyanathan, S., Kapitaniak, T.: A chaotic system with equilibria located on the rounded square loop and its circuit implementation. IEEE Trans. Circuits Syst. II Express Briefs 63(9), 878–882 (2016)CrossRefGoogle Scholar
  43. 43.
    Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations, vol. 99. Springer, Berlin (2013)MATHGoogle Scholar
  44. 44.
    Richard, J.-P.: Time-delay systems: an overview of some recent advances and open problems. Automatica 39(10), 1667–1694 (2003)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Cheng, C.-K., Kuo, H.-H., Hou, Y.-Y., Hwang, C.-C., Liao, T.-L.: Robust chaos synchronization of noise-perturbed chaotic systems with multiple time-delays. Physica A 387(13), 3093–3102 (2008)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Xiao, Y., Wei, X., Tang, S., Li, X.: Adaptive complete synchronization of the noise-perturbed two bi-directionally coupled chaotic systems with time-delay and unknown parametric mismatch. Appl. Math. Comput. 213(2), 538–547 (2009)MathSciNetMATHGoogle Scholar
  47. 47.
    He, W., Qian, F., Cao, J., Han, Q.-L.: Impulsive synchronization of two nonidentical chaotic systems with time-varying delay. Phys. Lett. A 375(3), 498–504 (2011)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Acho, L.: A continuous-time delay chaotic system obtained from a chaotic logistic map. http://www.actapress.com/Abstract.aspx?paperId=456387
  49. 49.
    Chai, Q.-Q.: A method of identifying parameters of a time-varying time-delay chaotic system. Acta Phys. Sin. 64(24), 0240506 (2015)MathSciNetGoogle Scholar
  50. 50.
    Tang, Y., Cui, M., Li, L., Peng, H., Guan, X.: Parameter identification of time-delay chaotic system using chaotic ant swarm. Chaos Solitons Fractals 41(4), 2097–2102 (2009)CrossRefGoogle Scholar
  51. 51.
    Liu, H., Yang, J.: Sliding-mode synchronization control for uncertain fractional-order chaotic systems with time delay. Entropy 17(6), 4202–4214 (2015)CrossRefGoogle Scholar
  52. 52.
    Tang, J.: Synchronization of different fractional order time-delay chaotic systems using active control. Math. Problems Eng. 2014, 262151 (2014).  https://doi.org/10.1155/2014/262151
  53. 53.
    He, S., Sun, K., Wang, H.: Synchronisation of fractional-order time delayed chaotic systems with ring connection. Eur. Phys. J. Spec. Topics 225(1), 97–106 (2016)CrossRefGoogle Scholar
  54. 54.
    Li, L., Peng, H., Yang, Y., Wang, X.: On the chaotic synchronization of Lorenz systems with time-varying lags. Chaos Solitons Fractals 41(2), 783–794 (2009)CrossRefMATHGoogle Scholar
  55. 55.
    Wang, S., Yu, Y.: Generalized synchronization of fractional order chaotic systems with time-delay. L2013 (2013)Google Scholar
  56. 56.
    Rajagopal, K., Karthikeyan, A., Duraisamy, P.: Hyperchaotic chameleon: fractional order FPGA implementation, complexity. 2017, 8979408 (2017).  https://doi.org/10.1155/2017/8979408
  57. 57.
    Jafari, M.A., Mliki, E., Akgul, A., Pham, V.-T., Kingni, S.T., Wang, X., Jafari, S.: Chameleon: the most hidden chaotic flow. Nonlinear Dyn. 88(3), 2303–2317 (2017)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Wei, Z., Sprott, J.C., Chen, H.: Elementary quadratic chaotic flows with a single non-hyperbolic equilibrium. Phys. Lett. A 379(37), 2184–2187 (2015)MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Li, C., Gong, Z., Qian, D., Chen, Y.Q.: On the bound of the Lyapunov exponents for the fractional differential systems. Chaos Interdiscip. J. Nonlinear Sci. 20(1), 013127 (2010)MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica D 16(3), 285–317 (1985)MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Ellner, S., Gallant, A.R., McCaffrey, D., Nychka, D.: Convergence rates and data requirements for Jacobian-based estimates of Lyapunov exponents from data. Phys. Lett. A 153(6–7), 357–363 (1991)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Maus, A., Sprott, J.C.: Evaluating lyapunov exponent spectra with neural networks. Chaos Solitons Fractals 51, 13–21 (2013)MathSciNetCrossRefMATHGoogle Scholar
  63. 63.
    Tavazoei, M.S., Haeri, M.: Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems. IET Signal Proc. 1(4), 171–181 (2007)CrossRefGoogle Scholar
  64. 64.
    Chudzik, A., Perlikowski, P., Stefanski, A., Kapitaniak, T.: Multistability and rare attractors in van der pol-duffing oscillator. Int. J. Bifurc. Chaos 21(07), 1907–1912 (2011)MathSciNetCrossRefMATHGoogle Scholar
  65. 65.
    Sprott, J.C., Wang, X., Chen, G.: Coexistence of point, periodic and strange attractors. Int. J. Bifurc. Chaos 23(05), 1350093 (2013)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Sprott, J.C., Li, C.: Asymmetric bistability in the Rössler system. Int. J. Bifurc. Chaos 48(1), 97 (2016)Google Scholar
  67. 67.
    Li, C., Hu, W., Sprott, J.C., Wang, X.: Multistability in symmetric chaotic systems. Eur. Phys. J. Spec. Topics 224(8), 1493–1506 (2015)CrossRefGoogle Scholar
  68. 68.
    Jaros, P., Borkowski, L., Witkowski, B., Czolczynski, K., Kapitaniak, T.: Multi-headed chimera states in coupled pendula. Eur. Phys. J. Spec. Topics 224(8), 1605–1617 (2015)CrossRefGoogle Scholar
  69. 69.
    Yuan, F., Wang, G., Wang, X.: Extreme multistability in a memristor-based multi-scroll hyper-chaotic system. Chaos Interdiscip. J. Nonlinear Sci. 26(7), 073107 (2016)MathSciNetCrossRefGoogle Scholar
  70. 70.
    Bao, B.C., Li, Q.D., Wang, N., Xu, Q.: Multistability in Chua’s circuit with two stable node-foci. Chaos Interdiscip. J. Nonlinear Sci. 26(4), 043111 (2016)MathSciNetCrossRefGoogle Scholar
  71. 71.
    Bao, B.C., Bao, H., Wang, N., Chen, M., Xu, Q.: Hidden extreme multistability in memristive hyperchaotic system. Chaos Solitons Fractals 94, 102–111 (2017)MathSciNetCrossRefMATHGoogle Scholar
  72. 72.
    Rajagopal, K., Akgul, A., Jafari, S., Karthikeyan, A., Koyuncu, I.: Chaotic chameleon: dynamic analyses, circuit implementation, FPGA design and fractional-order form with basic analyses. Chaos Solitons Fractals 103, 476–487 (2017)MathSciNetCrossRefGoogle Scholar
  73. 73.
    Akgul, A., Li, C., Pehlivan, I.: Amplitude control analysis of a four-wing chaotic attractor, its electronic circuit designs and microcontroller-based random number generator. J. Circuit Syst. Comput. 26(12), 1750190 (2017)CrossRefGoogle Scholar
  74. 74.
    Li, C., Sprott, J.C., Akgul, A., Iu, H.H.C., Zhao, Y.: A new chaotic oscillator with free control. Chaos Interdiscip. J. Nonlinear Sci. 27(8), 083101 (2017)MathSciNetCrossRefGoogle Scholar
  75. 75.
    Wen, H., Akgul, A., Li, C., Zheng, T., Li, P.: A switchable chaotic oscillator with two amplitude-frequency controllers. J. Circuit Syst. Comput. 26(10), 1750158 (2017)CrossRefGoogle Scholar
  76. 76.
    Pham, V.-T., Akgul, A., Volos, C., Jafari, S., Kapitaniak, T.: Dynamics and circuit realization of a no-equilibrium chaotic system with a boostable variable. AEU Int. J. Electron. Commun. 78, 134–140 (2017)CrossRefGoogle Scholar
  77. 77.
    Coskun, S., Tuncel, S., Pehlivan, I., Akgul, A.: Microcontroller-controlled electronic circuit for fast modelling of chaotic equations. Electron. World 121(1947), 24–25 (2015)Google Scholar
  78. 78.
    Akgul, A.: An electronic card for easy circuit realisation of complex nonlinear systems. Electron. World 124(1978), 29–31 (2017)Google Scholar
  79. 79.
    Banerjee, T., Biswas, D., Sarkar, B.C.: Design and analysis of a first order time-delayed chaotic system. Nonlinear Dyn. 70(1), 721–734 (2012)MathSciNetCrossRefGoogle Scholar
  80. 80.
    Valli, D., Muthuswamy, B., Banerjee, S., Ariffin, M.R.K., Wahab, A.W.A., Ganesan, K., Subramaniam, C.K., Kurths, J.: Synchronization in coupled Ikeda delay systems. Eur. Phys. J. Spec. Topics 223(8), 1465–1479 (2014)CrossRefGoogle Scholar
  81. 81.
    Tlelo-Cuautle, E., Pano-Azucena, A.D., Rangel-Magdaleno, J.J., Carbajal-Gomez, V.H., Rodriguez-Gomez, G.: Generating a 50-scroll chaotic attractor at 66 mHz by using FPGAS. Nonlinear Dyn. 85(4), 2143–2157 (2016)CrossRefGoogle Scholar
  82. 82.
    Wang, Q., Yu, S., Li, C., Lü, J., Fang, X., Guyeux, C., Bahi, J.M.: Theoretical design and FPGA-based implementation of higher-dimensional digital chaotic systems. IEEE Trans. Circuits Syst. I Regul. Pap. 63(3), 401–412 (2016)MathSciNetCrossRefGoogle Scholar
  83. 83.
    Dong, E., Liang, Z., Shengzhi, D., Chen, Z.: Topological horseshoe analysis on a four-wing chaotic attractor and its FPGA implement. Nonlinear Dyn. 83(1–2), 623–630 (2016)MathSciNetCrossRefGoogle Scholar
  84. 84.
    Tlelo-Cuautle, E., Carbajal-Gomez, V.H., Obeso-Rodelo, P.J., Rangel-Magdaleno, J.J., Nuñez-Perez, J.C.: FPGA realization of a chaotic communication system applied to image processing. Nonlinear Dyn. 82(4), 1879–1892 (2015)MathSciNetCrossRefGoogle Scholar
  85. 85.
    Rashtchi, V., Nourazar, M.: FPGA implementation of a real-time weak signal detector using a duffing oscillator. Circuits Syst. Signal Process. 34(10), 3101–3119 (2015)CrossRefGoogle Scholar
  86. 86.
    Tlelo-Cuautle, E., Rangel-Magdaleno, J.J., Pano-Azucena, A.D., Obeso-Rodelo, P.J., Nuñez-Perez, J.C.: FPGA realization of multi-scroll chaotic oscillators. Commun. Nonlinear Sci. Numer. Simul. 27(1), 66–80 (2015)MathSciNetCrossRefGoogle Scholar
  87. 87.
    Xu, Y.-M., Wang, L.-D., Duan, S.-K.: A memristor-based chaotic system and its field programmable gate array implementation. Acta Phys. Sin. 65(12), 120503 (2016).  https://doi.org/10.7498/aps.65.120503 Google Scholar
  88. 88.
    Rajagopal, K., Guessas, L., Karthikeyan, A., Srinivasan, A., Adam, G.: Fractional order memristor no equilibrium chaotic system with its adaptive sliding mode synchronization and genetically optimized fractional order PID synchronization. Complexity 2017, 1892618 (2017).  https://doi.org/10.1155/2017/1892618 MathSciNetMATHGoogle Scholar
  89. 89.
    Rajagopal, K., Karthikeyan, A., Srinivasan, A.K.: FPGA implementation of novel fractional-order chaotic systems with two equilibriums and no equilibrium and its adaptive sliding mode synchronization. Nonlinear Dyn. 87(4), 2281–2304 (2017)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Centre for Non-linear DynamicsDefense UniversityBishoftuEthiopia
  2. 2.Biomedical Engineering DepartmentAmirkabir University of TechnologyTehranIran
  3. 3.Department of Electrical and Electronics Engineering, Faculty of TechnologySakarya UniversitySakaryaTurkey
  4. 4.Department of Electronics EngineeringChennai Institute of TechnologyChennaiIndia

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