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

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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.

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References

  1. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)

    Article  MATH  Google Scholar 

  2. Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57(5), 397–398 (1976)

    Article  MATH  Google Scholar 

  3. Rössler, O.E.: Continuous chaos—four prototype equations. Ann. N. Y. Acad. Sci. 316(1), 376–392 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  4. Sprott, J.C.: Some simple chaotic flows. Phys. Rev. E 50(2), R647 (1994)

    Article  MathSciNet  Google Scholar 

  5. Schot, S.H.: Jerk: the time rate of change of acceleration. Am. J. Phys. 46(11), 1090–1094 (1978)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Sprott, J.C.: Some simple chaotic jerk functions. Am. J. Phys. 65(6), 537–543 (1997)

    Article  Google Scholar 

  10. Sprott, J.C.: Simplest dissipative chaotic flow. Phys. Lett. A 228(4–5), 271–274 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Munmuangsaen, B., Srisuchinwong, B.: Elementary chaotic snap flows. Chaos Solitons Fractals 44(11), 995–1003 (2011)

    Article  Google Scholar 

  12. Munmuangsaen, B., Srisuchinwong, B., Sprott, J.C.: Generalization of the simplest autonomous chaotic system. Phys. Lett. A 375(12), 1445–1450 (2011)

    Article  MATH  Google Scholar 

  13. Sprott, J.C.: A new chaotic jerk circuit. IEEE Trans. Circuits Syst. II Express Briefs 58(4), 240–243 (2011)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  20. Srisuchinwong, B., Nopchinda, D.: Current-tunable chaotic jerk oscillator. Electron. Lett. 49(9), 587–589 (2013)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  22. Leonov, G.A., Kuznetsov, N.V., Vagaitsev, V.I.: Localization of hidden chuas attractors. Phys. Lett. A 375(23), 2230–2233 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Leonov, G.A., Kuznetsov, N.V., Vagaitsev, V.I.: Hidden attractor in smooth chua systems. Physica D 241(18), 1482–1486 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  40. Molaie, M., Jafari, S., Sprott, J.C.: Simple chaotic flows with one stable equilibrium. Int. J. Bifurc. Chaos 23(11), 1350188 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  43. Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations, vol. 99. Springer, Berlin (2013)

    MATH  Google Scholar 

  44. Richard, J.-P.: Time-delay systems: an overview of some recent advances and open problems. Automatica 39(10), 1667–1694 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  48. Acho, L.: A continuous-time delay chaotic system obtained from a chaotic logistic map. http://www.actapress.com/Abstract.aspx?paperId=456387

  49. Chai, Q.-Q.: A method of identifying parameters of a time-varying time-delay chaotic system. Acta Phys. Sin. 64(24), 0240506 (2015)

    Google Scholar 

  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)

    Article  Google Scholar 

  51. Liu, H., Yang, J.: Sliding-mode synchronization control for uncertain fractional-order chaotic systems with time delay. Entropy 17(6), 4202–4214 (2015)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  55. Wang, S., Yu, Y.: Generalized synchronization of fractional order chaotic systems with time-delay. L2013 (2013)

  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. 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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  62. Maus, A., Sprott, J.C.: Evaluating lyapunov exponent spectra with neural networks. Chaos Solitons Fractals 51, 13–21 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  65. Sprott, J.C., Wang, X., Chen, G.: Coexistence of point, periodic and strange attractors. Int. J. Bifurc. Chaos 23(05), 1350093 (2013)

    Article  MathSciNet  Google Scholar 

  66. Sprott, J.C., Li, C.: Asymmetric bistability in the Rössler system. Int. J. Bifurc. Chaos 48(1), 97 (2016)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. Akgul, A.: An electronic card for easy circuit realisation of complex nonlinear systems. Electron. World 124(1978), 29–31 (2017)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. 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

    MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

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Authors and Affiliations

  1. Centre for Non-linear Dynamics, Defense University, Bishoftu, Ethiopia

    Karthikeyan Rajagopal

  2. Biomedical Engineering Department, Amirkabir University of Technology, Tehran, Iran

    Sajad Jafari

  3. Department of Electrical and Electronics Engineering, Faculty of Technology, Sakarya University, Sakarya, Turkey

    Akif Akgul

  4. Department of Electronics Engineering, Chennai Institute of Technology, Chennai, India

    Anitha Karthikeyan

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  1. Karthikeyan Rajagopal
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  2. Sajad Jafari
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Rajagopal, K., Jafari, S., Akgul, A. et al. Modified jerk system with self-exciting and hidden flows and the effect of time delays on existence of multi-stability. Nonlinear Dyn 93, 1087–1108 (2018). https://doi.org/10.1007/s11071-018-4247-5

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