Advertisement

Nonlinear Behavior in Fractional-Order Romeo and Juliet’s Love Model Influenced by External Force with Fuzzy Function

  • Linyun Huang
  • Youngchul BaeEmail author
Article
  • 11 Downloads

Abstract

Over the last three decades, chaotic dynamics in the field of mathematics, physics, chemistry, engineering, and social science have been studied by many researchers and scholars. In this paper, we reviewed period and chaotic behaviors in fractional-order love model of Romeo and Juliet with Gaussian fuzzy membership function as an external force or external environment. To do this, we fixed parameters a = − 1, b = − 5, c = d = 1 and varied α and β in the fractional-order differential equation of Romeo and Juliet love model, with α and β indicating the magnitude of love status of Romeo and Juliet, respectively. If α and β are the same, the magnitude of love status of Romeo is equal to that of Juliet. If α and β are different, then the magnitude of love status of Romeo is different from that of Juliet. We confirmed that when the difference of fractional order between Romeo and Juliet was small, their love status showed chaotic dynamics. This means that they love each other and the relationship between Romeo and Juliet is mutual and equivalent. When the difference is large, love status showed periodic motion. This means that the relationship between Romeo and Juliet is not equivalent.

Keywords

Fuzzy membership function Gaussian Fractional order Parameter Love model Fractional order 

References

  1. 1.
    Bae, Y.: Chaotic phenomena in addiction model for digital leisure. Int. J. Fuzzy Logic Intell. Syst. 13(4), 291–297 (2013).  https://doi.org/10.5391/IJFIS.2013.13.4.291 CrossRefGoogle Scholar
  2. 2.
    Bae, Y.: Chaotic dynamics in tobacco’s addiction model. Int. J. Fuzzy Logic Intell. Syst. 14(4), 322–331 (2014).  https://doi.org/10.5391/IJFIS.2014.14.4.322 CrossRefGoogle Scholar
  3. 3.
    Sprott, J.C.: Dynamical models of happiness. Nonlinear Dyn. Psychol. Life Sci. 9(1), 23–34 (2005)Google Scholar
  4. 4.
    Sprott, J.C.: Dynamics of Love and Happiness. In: Chaos and Complex Systems Seminar, Madison, WI, 2001Google Scholar
  5. 5.
    Bae, Y.: Synchronization of dynamical happiness model. Int. J. Fuzzy Logic Intell. Syst. 14(2), 91–97 (2014).  https://doi.org/10.5391/IJFIS.2014.14.2.91 CrossRefGoogle Scholar
  6. 6.
    Huang, L.Y., Bae, Y.: Analysis of nonlinear dynamics in family model including parent-in law. J Korean Inst. Intell. Syst. 26(1), 37–43 (2016).  https://doi.org/10.5391/jkiis.2016.26.1.037 CrossRefGoogle Scholar
  7. 7.
    Strogatz, S.H.: Love affairs and differential equations. Math. Mag. 61, 35 (1988).  https://doi.org/10.2307/2690328 MathSciNetCrossRefGoogle Scholar
  8. 8.
    Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Application to Physics, Biology, Chemistry and Engineering. Addison-Wesley, Reading (1994)Google Scholar
  9. 9.
    Bae, Y.: Chaotic behavior in dynamic love model with different external force. Int. J. Fuzzy Logic Intell. Syst. 15(4), 283–288 (2015).  https://doi.org/10.5391/IJFIS.2015.15.4.283 CrossRefGoogle Scholar
  10. 10.
    Bae, Y.: Nonlinear behavior in love model with discontinuous external force. Int. J. Fuzzy Logic Intell. Syst. 16(1), 64–71 (2016).  https://doi.org/10.5391/IJFIS.2016.16.1.64 CrossRefGoogle Scholar
  11. 11.
    Mandelbort, B.B.: The Fractal Geometry of Nature. Freeman, New York (1983)CrossRefGoogle Scholar
  12. 12.
    Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Piscataway (2001)zbMATHGoogle Scholar
  13. 13.
    Sun, H.H., Abdelwahad, A.A., Onaral, B.: Linear approximation of transfer function with a pole of fractional order. IEEE Trans. Auto Control 29, 4–441 (1984).  https://doi.org/10.1109/tac.1984.1103551 CrossRefGoogle Scholar
  14. 14.
    Bagley, R.L., Calico, R.A.: Fractional order state equations for the control of viscoelastically damped structures. Guid. Control Dyn. 14, 11–304 (1991).  https://doi.org/10.2514/3.20641 CrossRefGoogle Scholar
  15. 15.
    Kusnezov, D., Bulgac, A.A., Dang, G.D.: Quantum levy processes and fractional kinetics. Phys. Rev. Lett. 9, 1136 (1999).  https://doi.org/10.1103/physrevlett.82.1136 CrossRefGoogle Scholar
  16. 16.
    Wang, W., Chen, C.: Intelligent chaos synchronization of fractional order systems via mean-based slide mode controller. Int. J. Fuzzy Syst. 17(2), 144–157 (2015).  https://doi.org/10.1007/s40815-015-0030-7 MathSciNetCrossRefGoogle Scholar
  17. 17.
    Rinaldi, S.: Laura and Patriarch: an intriguing case of cyclical love dynamics. SIAM J. Appl. Math. 58, 1205–1221 (1998).  https://doi.org/10.1137/S003613999630592X MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cresswell, C.: Mathematics and Sex. Griffin Press, Sydney (2003)Google Scholar
  19. 19.
    Wauer, J., Schwarzer, D., Cai, G.Q., Lin, Y.K.: Dynamical models of love with time-varying fluctuations. Appl. Math. Comput. 188(2), 1535–1548 (2007).  https://doi.org/10.1016/j.amc.2006.11.026 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rinaldi, S.: Love dynamics: the case of linear couples. Appl. Math. Comput. 95(2–3), 181–192 (1998).  https://doi.org/10.1016/S0096-3003(97)10081-9 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Liao, X., Ran, J.: Hopf bifurcation in love dynamical models with nonlinear couples and time delays. Chaos Solitons Fractals 31(4), 853–865 (2007).  https://doi.org/10.1016/j.chaos.2005.10.037 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Breitenecker, F., Judex, F., Popper, N., Breitenecker, A.: Love emotions between Laura and Petrarch: an approach by mathematics and system dynamics. J. Comput. Inf. Technol. 16(4), 255–269 (2008).  https://doi.org/10.2498/cit.1001393 CrossRefGoogle Scholar
  23. 23.
    Huang, L.Y., Huang, S., Bae, Y.: Chaotic behavior in model with a Gaussian function as external force. Int. J. Fuzzy Logic Intell. Syst. 16(4), 262–269 (2016).  https://doi.org/10.5391/IJFIS.2016.16.4.262 CrossRefGoogle Scholar
  24. 24.
    Huang, L.Y., Bae, Y.: Nonlinear behavior in Romeo and Juliet’s love model influenced by external force with fuzzy membership function. Int. J. Fuzzy Syst. 19(2), 1670–1682 (2017).  https://doi.org/10.1007/s40815-017-0346-6 MathSciNetCrossRefGoogle Scholar
  25. 25.
    Huang, L.Y., Bae, Y.: Periodic doubling and chaotic attractor in the love model with a fourier series function as external force. Int. J. Fuzzy Logic Intell. Syst. 17(1), 17–2588 (2017).  https://doi.org/10.5391/IJFIS.2017.17.1.17 CrossRefGoogle Scholar
  26. 26.
    Ahmad, W.M., El-Khazali, R.: Fractional-order dynamical model of love. Chaos Solitons Fractals 33(4), 1367–1375 (2007).  https://doi.org/10.1016/j.chaos.2006.01.098 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ahmad, W.L., Chen, K.: Chaotic behavior in a new fractional-order love triangle system with competition. Appl. Anal. Comput. 5(1), 103–113 (2015).  https://doi.org/10.11948/2015009 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Huang, L.Y., Bae, Y.: Chaotic behavior in love affairs of fractional order with fuzzy membership function as an external force. In: International Journal of Fuzzy Systems Association and 9th International Conference on Soft Computing and Intelligent Systems, pp. 27–30 (2017).  https://doi.org/10.1109/ifsa-scis.2017.8023338
  29. 29.
    Huang, L.Y., Bae, Y.: Chaotic dynamics of the fractional-love model with external environment. Entropy 20(1), 53 (2018).  https://doi.org/10.3390/e20010053 MathSciNetCrossRefGoogle Scholar
  30. 30.
    Huang, L.Y., Bae, Y.: Analysis of chaotic behaviors in a novel extended love model considering positive and negative external environment. Entropy 20(5), 365 (2018).  https://doi.org/10.3390/e20050356 CrossRefGoogle Scholar
  31. 31.
    Yang, M., Wang, Q.: Approximate controllability of Riemann–Liouville fractional differential inclusions. Appl. Math. Comput. 274, 267–281 (2016).  https://doi.org/10.1016/j.amc.2015.11.017 MathSciNetCrossRefGoogle Scholar
  32. 32.
    Podlubny, I.: Fractional Differential Equations, pp. 159–290. Academic Press, New York (1999)zbMATHGoogle Scholar

Copyright information

© Taiwan Fuzzy Systems Association and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Biomedical and Electronic EngineeringChonnam National UniversityYeosuKorea
  2. 2.Division of Electrical, Electronic Communication and Computer EngineeringChonnam National UniversityYeosuKorea

Personalised recommendations