Chaos in the fractional order periodically forced complex Duffing’s oscillators. (English) Zbl 1088.37046

Summary: The occurrence of fractional-order chaotic dynamics have been intensively studied over the last ten years in a large number of real dynamical systems of physical nature. However, a similar study has not yet been carried out for fractional-order chaotic dynamical systems in the complex domain. In this paper, we numerically study the chaotic behaviors in the fractional-order symmetric and nonsymmetric periodically forced complex Duffing’s oscillators. We find that chaotic behaviors exist in the fractional-order periodically forced complex Duffing oscillators with orders less than 4. Our results are validated by the existence of positive maximal Lyapunov exponent.


37N05 Dynamical systems in classical and celestial mechanics
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
Full Text: DOI


[1] Hartley TT. The Duffing double scroll. In: Proc Amer Cont Conf. Pittsburgh, PA; June 1989. p. 419-23; Hartley TT. The Duffing double scroll. In: Proc Amer Cont Conf. Pittsburgh, PA; June 1989. p. 419-23
[2] Chiang, H.-D.; Liu, C.-W., Chaos in a simple power system, IEEE Trans Power Syst, 8, 4, 1407-1417 (1993)
[3] Stupnicka, W. S.; Rudowski, J., The coexistence of periodic, almost-periodic and chaotic attractors in the Van Der Pol-Duffing oscillator, J Sound Vib, 199, 2, 165-175 (1997) · Zbl 1235.70164
[4] Ajjarapu, V.; Lee, B., Bifurcation theory and its application to nonlinear dynamical phenomena in an electrical power system, IEEE Trans Power Syst, 7, 1, 424-431 (1992)
[5] Koeller, R. C., Application of fractional calculus to the theory of viscoelasticity, J Appl Mech, 51, June, 299 (1984) · Zbl 0544.73052
[6] Mandelbrot, B., Some noises with \(1/f\) spectrum, a bridge between direct current and white noise, IEEE Trans Inform Theory, IT-13, 2, 289-298 (1967) · Zbl 0148.40507
[7] Sun, H. H., Linear approximation of transfer function with a pole of fractional order, IEEE Trans Automat Contr, AC-29, 5, 441-444 (1984) · Zbl 0532.93025
[8] Podlubny, I., Fractional-order systems and \(PI^λ D^μ\)-controllers, IEEE Trans Automat Contr, 44, 1, 208-214 (1999) · Zbl 1056.93542
[9] Vinagre, B. M.; Petráš, I.; Podlubny, I.; Chen, Y. Q., Using fractional order adjustment rules and fractional order reference models in model-reference adaptive control, Nonlinear Dyn, 29, 269-279 (2002) · Zbl 1031.93110
[10] Hartley, T. T.; Lorenzo, C. F.; Killory Qammer, H., Chaos in a fractional order Chua’s system, IEEE Trans Circ Syst I: Fund Theory Appl, 42, 8, 485-490 (1995)
[11] Perrin, E.; Harba, R.; Berzin-Joseph, C.; Iribarren, I.; Bonami, A., \(n\) th-order fractional Brownian motion and fractional Gaussian noises, IEEE Trans Signal Process, 49, 5, 1049-1059 (2001)
[12] Ahmad, W.; Ei-Khazali, R.; Elwakil, A. S., Fractional order Wien-bridge oscillator, Electron. Lett., 37, 18/30, 1110-1112 (2001)
[13] Ahmad, W. M.; Sprott, J. C., Chaos in fractional-order autonomous nonlinear systems, Chaos, Solitons & Fractals, 16, 339-351 (2003) · Zbl 1033.37019
[14] Ahmad, W. M.; Harb, A. M., On nonlinear control design for autonomous chaotic systems of integer and fractional orders, Chaos, Solitons & Fractals, 18, 693-701 (2003) · Zbl 1073.93027
[15] Li, C.; Liao, X.; Yu, J., The synchronization of fractional-order systems, Phys Rev E, 68, 067203 (2003)
[16] Oldham, K. B.; Spanier, J., The fractional calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[17] Mahmoud, G. M.; Mohamed, A. A.; Aly, S. A., Strange attractors and chaos control in periodically forced complex Duffing’s oscillators, Physica A, 292, 193-206 (2001) · Zbl 0972.37054
[18] If, F.; Berg, P., Split-step spectral method for nonlinear Schrodinger equation with absorbing boundaries, J Comput Phys, 72, 501-503 (1987) · Zbl 0631.65129
[19] Hartley, T. T.; Lorenzo, C. F., Dynamics and control of initialized fractional-order systems, Nonlinear Dyn, 29, 201-233 (2002) · Zbl 1021.93019
[20] Charef, A.; Sun, H. H.; Tsao, Y. Y.; Onaral, B., Fractal system as represented by singularity function, IEEE Trans Autom Control, 37, Sept., 1465-1470 (1992) · Zbl 0825.58027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.