Abstract
The dynamical behavior on fractional-order Duffing system with two time scales is investigated, and the point-cycle coupling type cluster oscillation is firstly observed herein. When taking the fractional order as bifurcation parameter, the dynamics of the autonomous Duffing system will become more complex than the corresponding integer-order one, and some typical phenomenon exist only in the fractional-order one. Different attractors exist in various parameter space, and Hopf bifurcation only happens while fractional order is bigger than 1 under certain parameter condition. Moreover, the bifurcation behavior of the autonomous system may regulate dynamical phenomenon of the periodic excited system. It results into the point-cycle coupling type cluster oscillation when the fractional order is bigger than 1. The related generation mechanism based on slow-fast analysis method is that the slow variation of periodic excitation makes the system periodically visit different attractors and critical points of different bifurcations of the autonomous system.
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Pandey, V., Holm, S.: Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations. J. Acoust. Soc. Am. 140, 1–4 (2016)
Pinto, C., Carvalho, A.: A latency fractional order model for HIV dynamics. J. Comput. Appl. Math. 312, 240–256 (2017)
Sun, Y.F., Xiao, Y., Zheng, C.J., et al.: Modelling long-term deformation of granular soils incorporating the concept of fractional calculus. Acta. Mech. Sin. 32, 112–124 (2016)
Tarasova, V.V., Tarasov, V.E.: Concept of dynamic memory in economics. Commun. Nonlinear Sci. Numer. Simul. 55, 127–145 (2018)
Ying, Y.P., Lian, Y.P., Tang, S.Q., et al.: Enriched reproducing kernel particle method for fractional advection–diffusion equation. Acta. Mech. Sin. 34, 515–527 (2018)
Sun, Z.K., Liu, Y.Y., Liu, K., et al.: Aging transition in mixed active and inactive fractional-order oscillators. Chaos 29, 103–150 (2019)
Mansoori, M., Dehestani, M.: Anharmonic 1D actuator model including electrostatic and Casimir forces with fractional damping perturbed by an external force. Acta. Mech. Sin. 34, 528–541 (2018)
Niu, J.C., Shen, Y.J., Yang, S.P., et al.: Effect of a fractional-order PID controller on the dynamical response of a linear single degree-of-freedom oscillator. J. Vibr. Shock 35, 88–95 (2016)
Shen, Y.J., Wei, P., Yang, S.P.: Primary resonance of fractional-order van der Pol oscillator. Nonlinear Dyn. 77, 1629–1642 (2014)
Ge, Z.X., Chen, X.J., Hou, W.G.: Forced vibration resonance with fractional derivative damping. J. Appl. Math. A 30, 410–416 (2015)
Tan, X., Ding, H., Sun, J.Q., et al.: Primary and super-harmonic resonances of Timoshenko pipes conveying high-speed fluid. Ocean Eng. 203, 1–12 (2020)
Mao, X.Y., Sun, J.Q., Ding, H., et al.: An approximate method for one-dimensional structures with strong nonlinear and nonhomogeneous boundary conditions. J. Sound Vib. 469, 1–14 (2020)
Kheirizad, I., Tavazoei, M.S., Jalali, A.A.: Stability criteria for a class of fractional order systems. Nonlinear Dyn. 61, 153–161 (2010)
Mohammad, S.T., Mohammad, H.: A note on the stability of fractional order systems. Math. Comput. Simul. 79, 1566–1576 (2009)
Guo, P., Wang, Y.H., Tao, C.X., et al.: An approximate analytical solution to the family of time fractional Klein-Gordon equations. Sci. Technol. Innov. Herald 27, 47–52 (2009)
Zhu, X.G., Li, Y.X., Wu, B.: Stability analysis of fractional-order Langford systems. J. Shand. Univ. Sci. Technol. 38, 65–71 (2019)
Zhang, S.L.: Multilevel problem of fermentation process and its bioreactor device technology research fermentation process optimization and amplification technology based on process parameters. China Eng. Sci. 3, 37–45 (2001)
Sheintuch, M., Schmidt, J.: Bifurcations to periodic and aperiodic solutions during ammonia oxidation on a platinum wire. J. Phys. Chem. 92, 3404–3411 (1988)
Surana, A., Haller, G.: Ghost manifolds in slow-fast systems with applications to unsteady fluid flow separation. Physica D 237, 1507–1529 (2008)
Rinzel, J.: Bursting oscillations in an excitable membrane model. Lecture Notes Math. 1151, 304–316 (1985)
Han, X.J., Bi, Q.S., Kurths, J.: Route to bursting via pulse-shaped explosion. Phys. Rev. E 98, 1–5 (2018)
Wang, Q.Y., Murks, A., Perc, M., et al.: Taming desynchronized cluster with delays in the Macaque cortical network. Chin. Phys. B 20, 121–126 (2011)
Meng, P., Lu, Q.S., Zhao, Y., et al.: Dynamic analysis of synchronous cluster discharge in two compartment neuron model. J. Dyn. Control 14, 566–570 (2016)
Zhang, Z.D., Li, J., Liu, Y.N., et al.: The evolution mechanism of different forms of bursting oscillations in non-smooth dynamical systems. Sci. Sin. Technol. 49, 1031–1039 (2019)
Li, X.H., Hou, J.Y.: Bursting phenomenon in a piecewise mechanical system with parameter perturbation in stiffness. Int. J. Non-Linear Mech. 81, 165–176 (2016)
Hou, J., Yan, X.P., Li, P., et al.: Adaptive time-frequency representation for weak chirp signals based on Duffing oscillator stopping oscillation system. Int. J. Adapt. Control Signal Process. 32, 777–791 (2018)
Li, X.H., Shen, Y.J., Sun, J.Q., et al.: New periodic-chaotic attractors in slow-fast Duffing system with periodic parametric excitation. Sci. Rep. 9, 1–11 (2019)
Shi, J.F., Zhang, Y.L., Wang, L., et al.: Double-parameter bifurcation and global characteristic analysis of Duffing systems. Noise Vibr. Control 36, 32–37 (2016)
Zhang, Z.D., Peng, M., Qu, Z.F., et al.: Bursting oscillations and mechanism analysis in a non-smooth Duffing system with frequency domain of two time scales. Sci. Sin-Phys. Mech. Astron. 48, 22–33 (2018)
Li, X., Wu, R.C.: Hopf bifurcation analysis of a new commensurate fractional-order hyperchaotic system. Nonlinear Dyn. 78, 279–288 (2014)
Li, X.H., Tang, J.H., Wang, Y.L., et al.: Approximately analytical solution in slow-fast system based on modified multi-scale method. Appl. Math. Mech. 41, 605–622 (2020)
Han, X.J., Bi, Q.S., Zhang, C., et al.: Delayed bifurcations to repetitive spiking and classification of delay-induced bursting. Int. J. Bifurcat. Chaos 24, 1–23 (2014)
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grants 11672191, 11772206 and U1934201), and the Hundred Excellent Innovative Talents Support Program in Hebei University (Grant SLRC2017053).
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Wang, Y., Li, X. & Shen, Y. Cluster oscillation and bifurcation of fractional-order Duffing system with two time scales. Acta Mech. Sin. 36, 926–932 (2020). https://doi.org/10.1007/s10409-020-00967-y
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DOI: https://doi.org/10.1007/s10409-020-00967-y