Abstract
In this paper, we mainly study the bifurcation and resonance of under-damped Duffing systems with fractional order delay and fractional order under-damped Duffing systems with linear delay. Based on the separation method of fast and slow variables, the high-frequency excitation components in the system are eliminated, and the equivalent system of slow variables is obtained. For the under-damped Duffing system with fractional delay term, the harmonic balance method is used to solve the amplitude and phase analytic solution of the slow variable system, and for the under-damped fractional Duffing system with linear delay term, the average method is used to solve the amplitude and phase analytic solution of the slow variable system. Then the resonance and bifurcation of bistable and monostable systems with different parameters are analyzed. In the last section of this paper, numerical simulation is carried out to study the influence of fractional order, control parameters, delay quantity and other factors on the two systems, and the correctness of the analytical analysis is verified by comparing the numerical simulation results.
Similar content being viewed by others
Data availability
All data generated or analyzed during this study are included in this published article.
References
Shah, K., Abdeljawad, T., Ali, A.: Mathematical analysis of the Cauchy type dynamical system under piecewise equations with Caputo fractional derivative. Chaos Solitons Fractals 161, 112356 (2022)
Xie, J., Wang, H., Chen, L., et al.: Dynamical analysis of fractional oscillator system with cosine excitation utilizing the average method. Math. Method Appl. Sci. 45, 1–17 (2022)
Muñoz-Vázquez, A.J., Parra-Vega, V., Sánchez-Orta, A., et al.: High-gain fractional disturbance observer control of uncertain dynamical systems. J. Franklin Inst. 358(9), 4793–4806 (2021)
Duc, T.M., Van Hoa, N.: Stabilization of impulsive fractional-order dynamic systems involving the Caputo fractional derivative of variable-order via a linear feedback controller. Chaos Solitons Fractals 153, 111525 (2021)
Pishro, A., Shahrokhi, M., Sadeghi, H.: Fault-tolerant adaptive fractional controller design for incommensurate fractional-order nonlinear dynamic systems subject to input and output restrictions. Chaos Solitons Fractals 157, 111930 (2022)
Gu, S., He, S., Wang, H., et al.: Analysis of three types of initial offset-boosting behavior for a new fractional-order dynamical system. Chaos Solitons Fractals 143, 110613 (2021)
Shahnazi-Pour, A., Moghaddam, B.P., Babaei, A.: Numerical simulation of the Hurst index of solutions of fractional stochastic dynamical systems driven by fractional Brownian motion. J. Comput. Appl. Math. 386, 113210 (2021)
Razzaq, O.A., Khan, N.A., Faizan, M., et al.: Behavioral response of population on transmissibility and saturation incidence of deadly pandemic through fractional order dynamical system. Results Phys. 26, 104438 (2021)
Kumar, S., Kumar, R., Cattani, C., et al.: Chaotic behaviour of fractional predator–prey dynamical system. Chaos Solitons Fractals 135, 109811 (2020)
Dabiri, A., Moghaddam, B.P., Machado, J.A.T.: Optimal variable-order fractional PID controllers for dynamical systems. J. Comput. Appl. Math. 339, 40–48 (2018)
Moghaddam, B.P., Aghili, A.: A numerical method for solving linear non-homogenous fractional ordinary differential equation. Appl. Math. Inf. Sci. 6(3), 441–445 (2012)
Parsa Moghaddam, B., Dabiri, A., Mostaghim, Z.S., et al.: Numerical solution of fractional dynamical systems with impulsive effects. Int. J. Mod. Phys. C 34, 2350013 (2022)
Moghaddam, B.P., Dabiri, A., Machado, J.A.T.: Application of variable-order fractional calculus in solid mechanics. Appl. Eng. Life Soc. Sci. Part A 7, 207–224 (2019)
Arfan, M., Mahariq, I., Shah, K., et al.: Numerical computations and theoretical investigations of a dynamical system with fractional order derivative. Alex. Eng. J. 61(3), 1982–1994 (2022)
Shah, K., Arfan, M., Ullah, A., et al.: Computational study on the dynamics of fractional order differential equations with applications. Chaos Solitons Fractals 157, 111955 (2022)
Acay, B., Bas, E., Abdeljawad, T.: Non-local fractional calculus from different viewpoint generated by truncated M-derivative. J. Comput. Appl. Math. 366, 112410 (2020)
Ullah, N., Asjad, M.I.: Dynamics behavior of solitons solutions of Chen–Lee–Liu equation using analytical techniques. J. Fract. Calculus Nonlinear Syst. 3(1), 30–45 (2022)
Guo, Y., Li, Y.: Bipartite leader-following synchronization of fractional-order delayed multilayer signed networks by adaptive and impulsive controllers. Appl. Math. Comput. 430, 127243 (2022)
Wu, X., Liu, S., Wang, H.: Asymptotic stability and synchronization of fractional delayed memristive neural networks with algebraic constraints. Commun. Nonlinear Sci. 114, 106694 (2022)
Syam, M.I., Sharadga, M., Hashim, I.: A numerical method for solving fractional delay differential equations based on the operational matrix method. Chaos Solitons Fractals 147, 110977 (2021)
Shahmorad, S., Ostadzad, M.H., Baleanu, D.: A Tau–like numerical method for solving fractional delay integro–differential equations. Appl. Numer. Math. 151, 322–336 (2020)
Zhou, Y., Peng, L., Ahmad, B., et al.: Energy methods for fractional Navier-Stokes equations. Chaos Solitons Fractals 102, 78–85 (2017)
Alidousti, J., Eskandari, Z.: Dynamical behavior and Poincare section of fractional-order centrifugal governor system. Math. Comput. Simul. 182, 791–806 (2021)
Sun, S., Liu, L.: Multiple internal resonances in nonlinear vibrations of rotating thin-walled cylindrical shells. J. Sound Vib. 510, 116313 (2021)
Usama, B.I., Morfu, S., Marquie, P.: Vibrational resonance and ghost-vibrational resonance occurrence in Chua’s circuit models with specific nonlinearities. Chaos Solitons Fractals 153, 111515 (2021)
Sawkmie, I.S., Kharkongor, D.: Theoretical and numerical study of vibrational resonance in a damped softening Duffing oscillator. Int. J. Nonlinear Mech. 144, 104055 (2022)
Gui, R., Wang, Y., Yao, Y., et al.: Enhanced logical vibrational resonance in a two-well potential system. Chaos Solitons Fractals 138, 109952 (2020)
Usama, B.I., Morfu, S., Marquié, P.: Numerical analyses of the vibrational resonance occurrence in a nonlinear dissipative system. Chaos Solitons Fractals 127, 31–37 (2019)
Peng, J., Xiang, M., Wang, L., et al.: Nonlinear primary resonance in vibration control of cable-stayed beam with time delay feedback. Mech. Syst. Signal Process. 137, 106488 (2020)
Mbong, T.L.M.D., Siewe, M.S., Tchawoua, C.: Controllable parametric excitation effect on linear and nonlinear vibrational resonances in the dynamics of a buckled beam. Commun. Nonlinear Sci. 54, 377–388 (2018)
Ren, Y., Pan, Y., Duan, F.: Generalized energy detector for weak random signals via vibrational resonance. Phys. Lett. A 382(12), 806–810 (2018)
Yang, J.H., Zhu, H.: Bifurcation and resonance induced by fractional-order damping and time delay feedback in a Duffing system. Commun. Nonlinear Sci. 18(5), 1316–1326 (2013)
Rysak, A., Sedlmayr, M.: Damping efficiency of the Duffing system with additional fractional terms. Appl. Math. Model 111, 521–533 (2022)
Bezziou, M., Jebril, I., Dahmani, Z.: A new nonlinear duffing system with sequential fractional derivatives. Chaos Solitons Fractals 151, 111247 (2021)
Shangbin, J., Wei, J., Shuang, L., et al.: Research on detection method of multi-frequency weak signal based on stochastic resonance and chaos characteristics of Duffing system. Chin. J. Phys. 64, 333–347 (2020)
Nikolić, M., Rajković, M.: Bifurcations in nonlinear models of fluid-conveying pipes supported at both ends. J. Fluids Struct. 22(2), 173–195 (2006)
Niu, J., Liu, R., Shen, Y., et al.: Stability and bifurcation analysis of single-degree-of-freedom linear vibro-impact system with fractional-order derivative. Chaos Solitons Fractals 123, 14–23 (2019)
Holmes, P.J.: Bifurcations to divergence and flutter in flow-induced oscillations: a finite dimensional analysis. J. Sound Vib. 53(4), 471–503 (1977)
Xie, J., Zhao, F., He, D., et al.: Bifurcation and resonance of fractional cubic nonlinear system. Chaos Solitons Fractals 158, 112053 (2022)
Shang, H., Xu, J.: Delayed feedbacks to control the fractal erosion of safe basins in a parametrically excited system. Chaos Solitons Fractals 41(4), 1880–1896 (2009)
Chen, L., Basu, B., Nielsen, S.R.K.: Nonlinear periodic response analysis of mooring cables using harmonic balance method. J. Sound Vib. 438, 402–418 (2019)
Pei, L., Chong, A.S.E., Pavlovskaia, E., et al.: Computation of periodic orbits for piecewise linear oscillator by Harmonic Balance Methods. Commun. Nonlinear Sci. 108, 106220 (2022)
Bakirov, Z.B., Mikhailov, V.F.: Analysis of non-linear stochastic oscillations by the averaging method. J. Appl. Math. Mech. 78(5), 512–517 (2014)
Roy, R.V.: Averaging method for strongly non-linear oscillators with periodic excitations. Int. J. Nonlinear Mech. 29(5), 737–753 (1994)
Acknowledgements
This project is supported by National Natural Science Foundation of China (52005360, 52205404), Fundamental Research Program of Shanxi Province (202303021212293), and Scientific and Technological innovation Programs of Higher Education Institution in Shanxi (2021L403).
Author information
Authors and Affiliations
Contributions
The original authorship list: JX, RG, ZRn, DH, HX. The new authorship list: JX, RG, ZR, DH, HX. Since Dr. ZJ did not provide substantial contributions in the revised version, the revised version still retains the author order of the original submission, namely JX, RG, ZR, DH, HX. All authors have confirmed the above changes.
Corresponding authors
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Xie, J., Guo, R., Ren, Z. et al. Vibration resonance and fork bifurcation of under-damped Duffing system with fractional and linear delay terms. Nonlinear Dyn 111, 10981–10999 (2023). https://doi.org/10.1007/s11071-023-08462-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-023-08462-2