Abstract
For a class of generalized Duffing systems with two periodic excitations, the complex dynamical behaviors of the systems are investigated. In the first place, the bifurcation conditions and stability of the generalized Duffing system under single-period excitation are analyzed, and the bursting oscillations occurring under typical parameter conditions and their generation mechanisms are given. Then, the fast–slow analysis is applied for numerically simulating the system through time course diagrams and bifurcation diagrams, and the new bursting oscillation modes of the system with different parameters when the ratio of the frequencies of the two excitations is an integer multiple are studied, and the effects of single-excitation amplitude variations as well as double-excitation amplitude variations on the bursting oscillations are further investigated. Finally, by discretizing the system, the transition mechanism of single periodic attractor and chaotic attractor to various attractors are obtained.
Similar content being viewed by others
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
C. Kuehn, Multiple Time Scale Dynamics (Springer, New York, 2005)
H.F. Han, Q.S. Bi, Bursting oscillations as well as the mechanism in a filippov system with parametric and external excitations. Int. J. Bifurc. Chaos 30(12), 2050168 (2020)
M.K. Wei, W.A. Jiang, X.D. Ma, X.F. Zhang, X.J. Han, Q.S. Bi, Compound bursting dynamics in a parametrically and externally excited mechanical system. Chaos Solitons Fractals 143, 110605 (2021)
Y.T. Zhang, Q.J. Cao, W.H. Huang, Bursting oscillations in an isolation system with quasi-zero stiffness. Mech. Syst. Signal Process. 161, 107916 (2021)
L.X. Duan, T.T. Liang, Y.Q. Zhao, H.G. Xi, Multi-time scale dynamics of mixed depolarization block bursting. Nonlinear Dyn. 103(1), 1043–1053 (2021)
M.G. Pedersen, M. Brøns, M.P. Sørensen, Amplitude-modulated spiking as a novel route to bursting: Coupling-induced mixed-mode oscillations by symmetry breaking. Chaos 32(1), 013121 (2022)
Z. Rakaric, L.R. Lukesevic, On the phenomenon of bifurcation space symmetrization as mechanism for bursting oscillations generation. Arch. Appl. Mech. 93(2), 761–771 (2023)
L. Huang, G.Q. Wu, Z.D. Zhang, Q.S. Bi, Fast-slow dynamics and bifurcation mechanism in a novel chaotic system. Int. J. Bifurc. Chaos 29(10), 1930028 (2019)
Y. Lin, W.B. Liu, H. Bao, Q. Shen, Bifurcation mechanism of periodic bursting in a simple three-element-based memristive circuit with fast-slow effect. Chaos Solitons Fractals 131, 109524 (2020)
K.H.M. Nyman, P. Ashwin, P.D. Ditlevsen, Bifurcation of critical sets and relaxation oscillations in singular fast-slow systems. Nonlinearity 33(6), 2853–2904 (2020)
M.R. Zhang, X.F. Zhang, Q.S. Bi, Slow-fast behaviors and their mechanism in a periodically excited dynamical system with double hopf bifurcations. Int. J. Bifurc. Chaos 31(8), 2130022 (2021)
Y. Yu, W.Y. Zhou, Z.Y. Chen, Two fast/slow decompositions as well as period-adding sequences in a generalized Bonhoeffer-van der Pol electronic circuit. Int. J. Electron. Commun. 155, 154379 (2022)
E. Zhang, L. Yu, Z.Q. Yang, New topological classification of bursting in multi-time-scale Chay-Cook model. Eur. Phys. J. Special Topics 231(11–12), 2277–2288 (2022)
J. Rinzel, Bursting oscillations in an excitable membrane model. Ordinary Partial Differ. Equ. 1151(1), 304–316 (1985)
E.M. Izhikevich, Neural excitablity, spiking and bursting. Int. J. Bifurc. Chaos 10(6), 1171–1266 (2000)
B. Zhang, X.F. Zhang, X.D. Ma, Q.S. Bi, Influence mechanism of coexisted attractors on bursting oscillations. J. Vib. Shock 42(6), 224–231 (2023)
B. Zhang, X.F. Zhang, W.A. Jiang, H. Ding, L.Q. Chen, Q.S. Bi, Bursting oscillations induced by multiple coexisting attractors in a modified 3D van der Pol-Duffing system. Commun. Nonlinear Sci. Numer. Simul. 116, 106806 (2023)
X.J. Han, Q.S. Bi, Sliding fast–slow dynamics in the slowly forced Duffing system with frequency switching. Chaos Solitons Fractals 169, 113270 (2023)
K. Rajagopal, A.J.M. Khalaf, Z.C. Wei, V.T. Pham, A. Alsaedi, T. Hayat, Hyperchaos and coexisting attractors in a modified van der pol-duffing oscillator. Int. J. Bifurc. Chaos 29(5), 1950067 (2019)
X.D. Ma, Q.S. Bi, L.F. Wang, Complex bursting dynamics in the cubic-quintic Duffing-van der Pol system with two external periodic excitations. Meccanica 57(7), 1747–1766 (2022)
Z.Z. Zhang, X.D. Ma, Complex bursting oscillations as well as the mechanism of Duffing-van der Pol Oscillator with two weak periodic excitations. Jiangxi Sci. 40(2), 215–218 (2022)
X.F. Zhang, J.K. Zheng, G.Q. Wu, Q.S. Bi, Mixed mode oscillations as well as the bifurcation mechanism in a Duffing’s oscillator with two external periodic excitations. Sci. China Technol. Sci. 62(10), 1816–1824 (2019)
H.L. Wan, X.H. Li, Y.J. Shen, Y.L. Wang, Study on vibration reduction of dynamic vibration absorber for two-scale Duffing system. Chin. J. Theor. Appl. Mech. 54(11), 3136–3146 (2022)
X.J. Han, Y. Yu, C. Zhang, F.B. Xia, Q.S. Bi, Turnover of hysteresis determines novel bursting in Duffing system with multiple-frequency external forcings. Int. J. Non-Linear Mech. 89, 69–74 (2017)
C.Y. Zhou, F. Xie, Z.J. Li, Complex bursting patterns and fast-slow analysis in a smallest chemical reaction system with two slow parametric excitations. Chaos Solitons Fractals 137, 109859 (2020)
H.Q. Zhao, X.D. Ma, B. Zhang, Q.S. Bi, Bursting dynamics and the bifurcation mechanism of a modified Rayleigh-van der Pol-Duffing oscillator. Phys. Scr. 97(10), 105208 (2022)
Y.H. Qian, D.J. Zhang, Bursting oscillation and mechanism analysis of a class of Duffing-Van der Pol system with two excitation terms. Eur. Phys. J. Plus 138(11), 1017 (2023)
F. Zhao, X.D. Ma, S.Q. Cao, Periodic bursting oscillations in a hybrid Rayleigh-Van der Pol-Duffing oscillator. Nonlinear Dyn. 111(3), 2263–2279 (2023)
Acknowledgements
The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through grant No.12172333 and the Natural Science Foundation of Zhejiang through grant No.LY20A020003.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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
Qian, Y., Zhang, D. & Leng, M. Bursting dynamic analysis of generalized Duffing systems under two periodic excitations. Eur. Phys. J. Plus 139, 366 (2024). https://doi.org/10.1140/epjp/s13360-024-05178-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360-024-05178-z