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Non-smooth bifurcations on the bursting oscillations in a dynamic system with two timescales

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Abstract

The main purpose of the paper was to explore the structures of bursting oscillations as well as the mechanism of non-smooth dynamic system with two timescales. Based on the typical Chua’s system, a non-smooth dynamic system with two timescales is established, which can be considered as the coupling of slow and fast subsystems. Possible bifurcations as well as the critical conditions related to the equilibrium points of the fast subsystem in different regions divided by the switching boundaries are derived. The distribution of the eigenvalues related to the generalized Jacobian matrix reveals that multiple crossing bifurcations may occur at the switching boundaries, which may influence the dynamics of system. Periodic bursting oscillations can be observed, in which the fold-type non-smooth bifurcations connect the quiescent state and spiking state, leading to the symmetric focus/focus-fold/fold bursting attractors. Different behaviors can be observed at the intersection points between the trajectories and the switching boundaries, which can be explained by the characteristics of the equilibrium points in two regions beside the boundaries. Furthermore, it is found that the real parts of the pair of complex conjugate eigenvalues related to the foci may dramatically depend on certain parameters, which is employed to account for the change in the oscillation patterns related to the spiking state.

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References

  1. Hetzler, H.: On the effect of nonsmooth Coulomb friction on Hopf bifurcations in a 1-DoF oscillator with self-excitation due to negative damping. Nonlinear Dyn. 69, 601–614 (2012)

    Article  MathSciNet  Google Scholar 

  2. Leine, R.I.: Bifurcations of equilibria in non-smooth continuous systems. Phys. D 223, 121–137 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bernardo, M., Nordmark, A., Olivar, G.: Discontinuity-induced bifurcations of equilibria in piecewise-smooth and impacting dynamical systems. Phys. D 237, 119–136 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Angulo, F., Olivar, G., Osorio, G.A., Escobar, C.M., Ferreira, J.D., Redondo, J.M.: Bifurcations of non-smooth systems. Commun. Nonlinear Sci. Numer. Simul. 17, 4683–4689 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Hosseini, S.A.A., Zamanian, M.: Analytical solution for general nonlinear continuous systems in a complex form. Appl. Math. Model. 37, 1163–1169 (2013)

    Article  MathSciNet  Google Scholar 

  6. Tlelo-Cuautle, E., Muñoz-Pacheco, J.M.: Simulation of Chuas circuit by automatic control of step-size. Appl. Math. Comput. 190, 1526–1533 (2007)

    Article  MATH  Google Scholar 

  7. Leine, R.I.: Non-smooth stability analysis of the parametrically excited impact oscillator. Int. J. Non Linear Mech. 47, 1020–1032 (2012)

    Article  Google Scholar 

  8. Zhang, Z., Bi, Q.: Bifurcation in a piecewise linear circuit with switching boundaries. Int. J. Bifurc. Chaos 22, 12500341–125003418 (2012)

    Google Scholar 

  9. Simpson, D.J.W., Meiss, J.D.: Aspects of bifurcation theory for piecewise-smooth continuous systems. Phys. D 241, 1861–1868 (2012)

    Article  MathSciNet  Google Scholar 

  10. Ji, Y., Bi, Q.: Bursting behavior in a non-smooth electric circuit. Phys. Lett. A 374, 1434C1439 (2010)

    Google Scholar 

  11. Makarenkov, O., Lamb, J.S.W.: Dynamics and bifurcations of nonsmooth systems: a survey. Phys. D 241, 1826–1844 (2012)

    Article  MathSciNet  Google Scholar 

  12. Colombo, A., Bernardo, M., Hogan, S.J., Jeffrey, M.R.: Bifurcations of piecewise smooth flows: perspectives, methodologies and open problems. Phys. D 241, 1845–1860 (2012)

    Article  MathSciNet  Google Scholar 

  13. Teixeira, M.A., Silva, P.R.: Regularization and singular perturbation techniques for non-smooth systems. Phys. D 241, 1948–1955 (2012)

  14. Biemond, J.J., Wouw, N., Nijmeijer, H.: Bifurcations of equilibrium sets in mechanical systems with dry friction. Phys. D 241, 1882–1894 (2012)

    Article  MathSciNet  Google Scholar 

  15. Feckan, M., Pospsil, M.: Bifurcation of sliding periodic orbits in periodically forced discontinuous systems. Nonlinear Anal. Real World Appl. 14, 150–162 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kingni, S.T., Keuninckx, L., Woafo, P., Van der Sande, G., Danckaert, J.: Dissipative chaos, Shilnikov chaos and bursting oscillations in a three-dimensional autonomous system: theory and electronic implementation. Nonlinear Dyn. 73, 1111–1123 (2013)

    Article  MATH  Google Scholar 

  17. 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  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Zhang, Z., Li, Y., Bi, Q.: Routes to bursting in a periodically driven oscillator. Phys. Lett. A 377, 975–980 (2013)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The authors are supported by the National Natural Science Foundation of China (11272135, 21276115) and Qing Lan Project of Jiangsu Province.

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Authors and Affiliations

  1. Faculty of Science, Jiangsu University, Zhenjiang, 212013, China

    Zhengdi Zhang, Binbin Liu & Qinsheng Bi

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  1. Zhengdi Zhang
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  2. Binbin Liu
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  3. Qinsheng Bi
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Correspondence to Qinsheng Bi.

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Zhang, Z., Liu, B. & Bi, Q. Non-smooth bifurcations on the bursting oscillations in a dynamic system with two timescales. Nonlinear Dyn 79, 195–203 (2015). https://doi.org/10.1007/s11071-014-1656-y

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  • DOI: https://doi.org/10.1007/s11071-014-1656-y

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