Fractional-order dynamical models of love. (English) Zbl 1133.91539

Ahmad, Wajdi M.; El-Khazali, Reyad

Chaos Solitons Fractals 33, No. 4, 1367-1375 (2007).

Summary: This paper examines fractional-order dynamical models of love. It offers a generalization of a dynamical model recently reported in the literature. The generalization is obtained by permitting the state dynamics of the model to assume fractional orders. The fact that fractional systems possess memory justifies this generalization, as the time evolution of romantic relationships is naturally impacted by memory. We show that with appropriate model parameters, strange chaotic attractors may be obtained under different fractional orders, thus confirming previously reported results obtained from integer-order models, yet at an advantage of reduced system order. Furthermore, this work opens a new direction of research whereby fractional derivative applications might offer more insight into the modeling of dynamical systems in psychology and life sciences. Our results are validated by numerical simulations.

MSC:

91D30	Social networks; opinion dynamics
26A33	Fractional derivatives and integrals
34C60	Qualitative investigation and simulation of ordinary differential equation models

Software:

Sprott's Software

References:

[1]	Arena, P.; Caponetto, R.; Fortuna, L.; Porto, D., Nonlinear non integer order circuits and systems—an introduction, (2000), World Scientific · Zbl 0966.93006
[2]	Podlubny, I., Fractional differential equations, (1999), Academic Press · Zbl 0918.34010
[3]	Gorenflo R, Mainardi F. Fractional calculus: integral and differential equations of fractional order. CISM course scaling laws and fractality in continuum mechanics, lecture notes, Udine, Italy, September 23-27, 1996.
[4]	Oldham, K.; Spanier, J., Fractional calculus, (1974), Academic Press NewYork · Zbl 0428.26004
[5]	Ahmad, W.; El-Khazali, R.; El-Wakil, A., Fractional wein-bridge oscillators, IEE electron lett, 37, 1110-1112, (2001)
[6]	Ahmad W, El-Khazali R, Fractional-order passive low-pass filters. In: International Conf. on electronics, circuits and systems, Sharjah, UAE, December 14-17, 2003, p. 160-3.
[7]	Ahmad W. Fractional capacitors for power factor correction. In: Proceedings IEEE Conf. on circuits and systems, ISCAS, Bangkok, Thailand, III-5-III-8, May 2003.
[8]	Ahmad W, El-Khazali R. On frequency response of fractional order Sallen-key filters. In: Proceedings of IFAC workshop on fractional differentiation and its application, France, July 2004, p. 571-4.
[9]	Ahmad, W.; Sprott, J.C., Chaos in fractional-order autonomous nonlinear systems, Chaos, solitons, & fractals, 16, 339-351, (2003) · Zbl 1033.37019
[10]	Ahmad, W.; Harb, A., On nonlinear control design for autonomous chaotic systems of integer and fractional orders, Chaos, solitons, & fractals, 18, 693-701, (2003) · Zbl 1073.93027
[11]	El-Khazali R, Ahmad W, Al-Assaf Y. Sliding mode control of fractional chaotic systems. In: Proceedings of IFAC workshop on fractional differentiation and its application, France, July 2004, p. 495-500. · Zbl 1139.93009
[12]	Ahmad, W.; El-Khazali, R.; Al-Assaf, Y., Stabilization of fractional chaotic systems using state-feedback control, Chaos, solitons, & fractals, 22, 1, 141-150, (2004) · Zbl 1060.93515
[13]	Sprott, J.C., Dynamical models of love, Nonlinear dyn. psychol, life sci, 8, 3, (2004)
[14]	Gorenflo, G., Fractional calculus: some numerical methods, (), 277-290