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Multi-bifurcation cascaded bursting oscillations and mechanism in a novel 3D non-autonomous circuit system with parametric and external excitation

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Abstract

In this paper, multi-timescale dynamics and the formation mechanism of a 3D non-autonomous system with two slowly varying periodic excitations are systematically investigated. Interestingly, the system shows novel multi-bifurcation cascaded bursting oscillations (MBCBOs) when the frequency of the two excitations is much lower than the mean frequency of the original system. For instance, periodic, quasi-periodic and chaotic bursting oscillations induced by a variety of cascaded bifurcations are first observed, and the phenomenon of spiking transfer is also revealed. Besides, stability and local bifurcations of the system are comprehensively investigated to analyze the mechanism of the observed MBCBOs, in which bifurcation diagram, Lyapunov exponents, time series, phase portraits, and transformed phase diagrams are used. Finally, through a circuit simulation and hardware experiment, these complex dynamics phenomena are verified physically.

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References

  1. Matzakos-Karvouniari, D., Gil, L., Orendorff, E., Marre, O., Picaud, S., Cessac, B.: A biophysical model explains the spontaneous bursting behavior in the developing retina. Sci. Rep. 9, 1859 (2019)

    Article  Google Scholar 

  2. Español, M.I., Rotstein, H.G.: Complex mixed-mode oscillatory patterns in a periodically forced excitable Belousov–Zhabotinsky reaction model. Chaos 25, 064612 (2015)

    Article  Google Scholar 

  3. Chen, Y., Tse, C.K., Qiu, S., Lindenmuller, L., Schwarz, W.: Coexisting fast-scale and slow-scale instability in current-mode controlled dc/dc converters: analysis, simulation and experimental results. IEEE Trans. Circuits Syst. I-Regul. Pap. 55, 3335–3348 (2008)

    Article  MathSciNet  Google Scholar 

  4. Li, H., Chen, D., Gao, X., Wang, X., Han, Q., Wu, C.: Fast-slow dynamics of a hydropower generation system with multi-time scales. Mech. Syst. Signal Proc. 110, 458–468 (2018)

    Article  Google Scholar 

  5. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci 20, 130–141 (1963)

    Article  MathSciNet  Google Scholar 

  6. Yu, S., Lu, J., Yu, X., Chen, G.: Design and implementation of grid multiwing hyperchaotic lorenz system family via switching control and constructing super-heteroclinic loops. IEEE Trans. Circuits Syst. I-Regul. Pap. 59, 1015–1028 (2012)

    Article  MathSciNet  Google Scholar 

  7. Cho, K., Miyano, T.: Chaotic cryptography using augmented Lorenz equations aided by quantum key distribution. IEEE Trans. Circuits Syst. I-Regul. Pap. 62, 478–487 (2015)

    Article  Google Scholar 

  8. Cang, S., Li, Y., Zhang, R., Wang, Z.: Hidden and self-excited coexisting attractors in a Lorenz-like system with two equilibrium points. Nonlinear Dyn. 95, 381–390 (2019)

    Article  Google Scholar 

  9. Izhikevich, E.M.: Neural excitability, spiking and bursting. Int. J. Bifurcation Chaos 10, 1171–1266 (2000)

    Article  MathSciNet  Google Scholar 

  10. Siewe, R.T., Domguia, U.S., Woafo, P.: Generation of pulse-like and bursting-like oscillations from nonlinear systems using embedded technologies and applications to excite mechanical arms. Commun. Nonlinear Sci. Numer. Simul. 69, 343–359 (2019)

    Article  Google Scholar 

  11. Duan, Q., Jing, Z., Zou, X., Wang, Y., Yang, K., Zhang, T., Wu, S., Huang, R., Yang, Y.: Spiking neurons with spatiotemporal dynamics and gain modulation for monolithically integrated memristive neural networks. Nat. Commun. 11, 3399 (2020)

    Article  Google Scholar 

  12. Huang, X., Ruan, X., Du, F., Liu, F., Zhang, L.: A pulsed power supply adopting active capacitor converter for low-voltage and low-frequency pulsed loads. IEEE Trans. Power Electron. 33, 9219–9230 (2018)

    Article  Google Scholar 

  13. Krahe, R., Gabbiani, F.: Burst firing in sensory systems. Nat. Rev. Neurosci. 5(1), 13–23 (2004)

    Article  Google Scholar 

  14. Bi, Q., Zhang, Z.: Bursting phenomena as well as the bifurcation mechanism in controlled Lorenz oscillator with two time scales. Phys. Lett. A 375, 1183–1190 (2011)

    Article  Google Scholar 

  15. Bao, H., Hu, A., Liu, W., Bao, B.: Hidden bursting firings and bifurcation mechanisms in memristive neuron model with threshold electromagnetic induction. IEEE Trans. Neural Netw. Learn. Syst. 31, 502–511 (2020)

    Article  Google Scholar 

  16. Izhikevich, E.M.: Dynamical systems in neuroscience: the geometry of excitability and bursting. MIT Press, Cambridge (2006)

    Book  Google Scholar 

  17. Han, X., Zhang, Y., Bi, Q., Kurths, J.: Two novel bursting patterns in the duffing system with multiple-frequency slow parametric excitations. Chaos 28, 043111 (2018)

    Article  MathSciNet  Google Scholar 

  18. Han, X., Bi, Q., Ji, P., Kurths, J.: Fast-slow analysis for parametrically and externally excited systems with two slow rationally related excitation frequencies. Phys. Rev. E 92, 012911 (2015)

    Article  MathSciNet  Google Scholar 

  19. Bertram, R., Rubin, J.E.: Multi-timescale systems and fast-slow analysis. Math. Biosci. 287, 105–121 (2017)

    Article  MathSciNet  Google Scholar 

  20. Krupa, M., Vidal, A., Desroches, M., et al.: Mixed-mode oscillations in a multiple time scale phantom bursting system. SIAM J. Appl. Dyn. Syst. 11(4), 1458–1498 (2012)

    Article  MathSciNet  Google Scholar 

  21. Cai, G., Tan, Z., Zhou, W., Tu, W.: Dynamical analysis of a new chaotic system and its chaotic control. Acta Phys. Sin. 56, 6230–6237 (2007)

    MATH  Google Scholar 

  22. Wang, M., Zeng, Y., Chen, G., He, J.: Nonresonant parametric control of Chen’s system. Acta Phys. Sin. 60, 010509 (2011)

  23. Park, E.-H., Zaks, M.A., Kurths, J.: Phase synchronization in the forced Lorenz system. Phys. Rev. E 60, 6627–6638 (1999)

    Article  MathSciNet  Google Scholar 

  24. Teka, W., Tabak, J., Bertram, R.: The relationship between two fast/slow analysis techniques for bursting oscillations. Chaos 22, 043117 (2012)

    Article  MathSciNet  Google Scholar 

  25. Han, X., Liu, Y., Bi, Q., Kurths, J.: Frequency-truncation fast-slow analysis for parametrically and externally excited systems with two slow incommensurate excitation frequencies. Commun. Nonlinear Sci. Numer. Simul. 72, 16–25 (2019)

    Article  MathSciNet  Google Scholar 

  26. Cho, E.: De moivre’s formula for quaternions. Appl. Math. Lett. 11, 33–35 (1998)

  27. Yu, S., Lu, J., Chen, G.: Theoretical design and circuit implementation of multidirectional multi-torus chaotic attractors. IEEE Trans. Circuits Syst. I-Regul. Pap. 54, 2087–2098 (2007)

    Article  MathSciNet  Google Scholar 

  28. Zhou, C., Li, Z., Xie, F., Ma, M., Zhang, Y.: Bursting oscillations in Sprott B system with multi-frequency slow excitations: two novel “Hopf/Hopf”-hysteresis-induced bursting and complex AMB rhythms. Nonlinear Dyn. 97, 2799–2811 (2019)

  29. DeJesus, E.X., Kaufman, C.: Routh–Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations. Phys. Rev. A 35, 5288–5290 (1987)

    Article  MathSciNet  Google Scholar 

  30. Buscarino, A., Fortuna, L., Frasca, M.: Essentials of Nonlinear Circuit Dynamics with MATLAB and Laboratory Experiments. CRC Press, Boca Raton (2017)

    Book  Google Scholar 

  31. Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics, Analytical, Computational, and Experimental Methods. Wiley, Chichester (1995)

    Book  Google Scholar 

Download references

Acknowledgements

This work was supported by the Research Foundation of Education Department of Hunan Province, China (Grant No. 20B567), the Natural Science Foundation of Hunan Province, China (Grant No. 2019JJ50624), and National Natural Science Foundation of China (Grant No. 62071411). This work was also funded by the China Scholarship Council (Grant No. 201808430258).

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Authors and Affiliations

  1. School of Automation and Electronic Information, Xiangtan University, Xiangtan, 411105, People’s Republic of China

    Mengjiao Wang, Jianhui Li & Zhijun Li

  2. School of Electrical, Electronic and Computer Engineering, University of Western Australia, Crawley, WA, 6009, Australia

    Mengjiao Wang, Xinan Zhang, Herbert Ho-Ching Iu & Tyrone Fernando

  3. School of Physics and Optoelectric Engineering, Xiangtan University, Xiangtan, 411105, People’s Republic of China

    Yicheng Zeng

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  1. Mengjiao Wang
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Correspondence to Mengjiao Wang.

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Wang, M., Li, J., Zhang, X. et al. Multi-bifurcation cascaded bursting oscillations and mechanism in a novel 3D non-autonomous circuit system with parametric and external excitation. Nonlinear Dyn 105, 3699–3714 (2021). https://doi.org/10.1007/s11071-021-06763-y

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