Skip to main content
SpringerLink
Log in

Forecasting bifurcations in parametrically excited systems

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Forecasting bifurcations in parametrically excited systems before they occur is an active area of research both for engineered and natural systems. In particular, anticipating the distance to critical transitions, and predicting the state of the system after such transitions, remains a challenge, especially when there is an explicit time input to the system. In this work, a new model-less method is presented to address these challenges based on monitoring transient recoveries from large perturbations in the pre-bifurcation regime. Recoveries are studied in a Poincaré section to address the challenge caused by explicit time input. Both numerical and experimental results are presented to demonstrate the proposed technique. A discussion of the accuracy of the proposed approach is included also.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Perkins, N.C.: Modal interactions in the non-linear response of elastic cables under parametric/external excitation. Int. J. Non-Linear Mech. 27(2), 233–250 (1992)

    Article  MATH  Google Scholar 

  2. Blankenship, G., Kahraman, A.: Steady state forced response of a mechanical oscillator with combined parametric excitation and clearance type non-linearity. J. Sound Vib. 185(5), 743–765 (1995)

    Article  MATH  Google Scholar 

  3. Bobin, J., Decroisette, M., Meyer, B., Vitel, Y.: Harmonic generation and parametric excitation of waves in a laser-created plasma. Phys. Rev. Lett. 30(13), 594 (1973)

    Article  Google Scholar 

  4. Panda, L., Kar, R.: Nonlinear dynamics of a pipe conveying pulsating fluid with parametric and internal resonances. Nonlinear Dyn. 49(1), 9–30 (2007)

    Article  MATH  Google Scholar 

  5. Belhaq, M., Houssni, M.: Quasi-periodic oscillations, chaos and suppression of chaos in a nonlinear oscillator driven by parametric and external excitations. Nonlinear Dyn. 18(1), 1–24 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Raghothama, A., Narayanan, S.: Periodic response and chaos in nonlinear systems with parametric excitation and time delay. Nonlinear Dyn. 27(4), 341–365 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (2008)

    MATH  Google Scholar 

  8. Taylor, A., Sherratt, J.A., White, A.: Seasonal forcing and multi-year cycles in interacting populations: lessons from a predator–prey model. J. Math. Biol. 67(6–7), 1741 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bolzoni, L., Dobson, A.P., Gatto, M., Leo, G.A.D.: Allometric scaling and seasonality in the epidemics of wildlife diseases. Am. Nat. 172(6), 818 (2008)

    Article  Google Scholar 

  10. Keeling, M.J., Rohani, P.: Modeling Infectious Diseases in Humans and Animals. Princeton University Press, Princeton (2008)

    MATH  Google Scholar 

  11. Lei, S., Zhang, W., Lin, J., Kennedy, Q.D., Williams, F.W.: Frequency domain response of a parametrically excited riser under random wave forces. J. Sound Vib. 333(2), 485 (2014)

    Article  Google Scholar 

  12. Oropeza-Ramos, A.L., Turner, L.K.: Parametric resonance amplification in a memgyroscope. In: Sensors, 2005 IEEE, p. 4, IEEE (2005)

  13. Bulian, G., Francescutto, A., Lugni, C.: On the nonlinear modeling of parametric rolling in regular and irregular waves. Int. Shipbuild. Prog. 51(2), 173–203 (2004)

    Google Scholar 

  14. Jia, Y., Yan, J., Soga, K., Seshia, A.A.: A parametrically excited vibration energy harvester. J. Intell. Mater. Syst. Struct. 25(3), 278–289 (2014)

    Article  Google Scholar 

  15. Hirsch, M.W., Smale, S., Devaney, R.L.: Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic press, Boca Raton (2012)

    MATH  Google Scholar 

  16. Sundararajan, P., Noahn, S.T.: Dynamics of forced nonlinear systems using shooting/arc-length continuation methodapplication to rotor systems. J. Vib. Acoust. 119(1), 9 (1997)

    Article  Google Scholar 

  17. Scheffer, M., Carpenter, S.R., Lenton, T.M., Bascompte, J., Brock, W., Dakos, V., Van De Koppel, j, Leemput, I .A.V .D., Leemput, S.A., Levin, E .H .V.Nes, et al.: Anticipating critical transitions. Science 338(6105), 344 (2012)

    Article  Google Scholar 

  18. Drake, J.M., Griffen, B.D.: Early warning signals of extinction in deteriorating environments. Nature 467(7314), 456 (2010)

    Article  Google Scholar 

  19. Lenton, T.M., Held, H., Kriegler, E., Hall, J.W., Lucht, W., Rahmstorf, S., Schellnhuber, H.J.: Tipping elements in the earth’s climate system. Proc. Natl. Acad. Sci. 105(6), 1786 (2008)

    Article  MATH  Google Scholar 

  20. Scheffer, M., Bascompte, J., Brock, W.A., Brovkin, V., Carpenter, S.R., Dakos, V., Held, H., van Nes, E.H., Rietkerk, M., Sugihara, G.: Early-warning signals for critical transitions. Nature 461(7260), 53 (2009)

    Article  Google Scholar 

  21. Kuehn, C.: A mathematical framework for critical transitions: bifurcations, fast–slow systems and stochastic dynamics. Phys. D Nonlinear Phenom. 240(12), 1020 (2011)

    Article  MATH  Google Scholar 

  22. Thompson, J.M.T., Sieber, J.: Predicting climate tipping as a noisy bifurcation: a review. Int. J. Bifurc. Chaos 21(02), 399 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dakos, V., Scheffer, M., van Nes, E.H., Brovkin, V., Petoukhov, V., Held, H.: Slowing down as an early warning signal for abrupt climate change. Proc. Natl. Acad. Sci. 105(38), 14308 (2008)

    Article  Google Scholar 

  24. Boettiger, C., Hastings, A.: Quantifying limits to detection of early warning for critical transitions. J. R. Soc. Interface 9(75), 2527 (2012)

    Article  Google Scholar 

  25. Lim, J., Epureanu, B.I.: Forecasting a class of bifurcations: theory and experiment. Phys. Rev. E 83(1), 016203 (2011)

    Article  MathSciNet  Google Scholar 

  26. Ghadami, A., Epureanu, B.I.: Bifurcation forecasting for large dimensional oscillatory systems: forecasting flutter using gust responses. J. Comput. Nonlinear Dyn. 11(6), 061009 (2016)

    Article  Google Scholar 

  27. Ghadami, A., Epureanu, B.I.: Forecasting the post-bifurcation dynamics of large-dimensional slow-oscillatory systems using critical slowing down and center space reduction. Nonlinear Dyn. 88(1), 415–431 (2017)

    Article  Google Scholar 

  28. Spina, D., Valente, C., Tomlinson, G.R.: A new procedure for detecting nonlinearity from transient data using the gabor transform. Nonlinear Dyn. 11(3), 235–254 (1996)

    Article  MathSciNet  Google Scholar 

  29. Kerschen, G., et al.: Past, present and future of nonlinear system identification in structural dynamics. Mech. Syst. Signal Process. 20(3), 505–592 (2006)

    Article  Google Scholar 

  30. Thompson, J.M.T., Virgin, L.N.: Predicting a jump to resonance using transient maps and beats. Int. J. Non-Linear Mech. 21(3), 205–216 (1986)

    Article  MathSciNet  Google Scholar 

  31. Lim, J., Epureanu, B.I.: Forecasting bifurcation morphing: application to cantilever-based sensing. Nonlinear Dyn. 67(3), 2291–2298 (2012)

    Article  Google Scholar 

  32. Guckenheimer, J., Philip J.H.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, vol. 42. Springer Science & Business Media (2013)

  33. May, R.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (1973)

    Google Scholar 

  34. Hallam, T., Clark, C.: Non-autonomous logistic equations as models of populations in a deteriorating environment. J. Theor. Biol. 93(2), 303–311 (1981)

    Article  Google Scholar 

Download references

Acknowledgements

This research was supported by the National Institute of General Medical Sciences of the National Institutes of Health under Award Number U01GM110744. The content is solely the responsibility of the authors and does not necessarily reflect the official views of the National Institutes of Health.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of Michigan, 2350 Hayward Street, Ann Arbor, MI, 48109, USA

    Shiyang Chen & Bogdan Epureanu

Authors
  1. Shiyang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bogdan Epureanu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bogdan Epureanu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, S., Epureanu, B. Forecasting bifurcations in parametrically excited systems. Nonlinear Dyn 91, 443–457 (2018). https://doi.org/10.1007/s11071-017-3880-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3880-8

Keywords

Search

Navigation