Nonlinear Dynamics

, Volume 93, Issue 2, pp 251–258 | Cite as

Predicting non-stationary and stochastic activation of saddle-node bifurcation in non-smooth dynamical systems

  • Jinki Kim
  • K. W. Wang
Original Paper


Saddle-node bifurcation can cause dynamical systems undergo large and sudden transitions in their response, which is very sensitive to stochastic and non-stationary influences that are unavoidable in practical applications. Therefore, it is essential to simultaneously consider these two factors for estimating critical system parameters that may trigger the sudden transition. Although many systems exhibit non-smooth dynamical behavior, estimating the onset of saddle-node bifurcation in them under the dual influence remains a challenge. In this work, a new theoretical framework is developed to provide an effective means for accurately predicting the probable time at which a non-smooth system undergoes saddle-node bifurcation while the governing parameters are swept in the presence of noise. The stochastic normal form of non-smooth saddle-node bifurcation is scaled to assess the influence of noise and non-stationary factors by employing a single parameter. The Fokker–Planck equation associated with the scaled normal form is then utilized to predict the distribution of the onset of bifurcations. Experimental efforts conducted using a double-well Duffing analog circuit successfully demonstrate that the theoretical framework developed in this study provides accurate prediction of the critical parameters that induce non-stationary and stochastic activation of saddle-node bifurcation in non-smooth dynamical systems.


Bifurcation Saddle-node Non-smooth Piecewise-linear Normal form Duffing bistable 



This research was partially supported by the National Science Foundation under Award Nos. 1232436 and 1661568 and the University of Michigan Collegiate Professorship fund.


  1. 1.
    Kuznetsov, Y.: Elements of Applied Bifurcation Theory, 2nd edn. Springer, New York (1998)zbMATHGoogle Scholar
  2. 2.
    Kovacic, I., Brennan, M.: The Duffing Equation: Nonlinear Oscillators and Their Behaviour. Wiley, Hoboken (2011)CrossRefzbMATHGoogle Scholar
  3. 3.
    Devoret, M., Esteve, D., Martinis, J., Cleland, A., Clarke, J.: Resonant activation of a Brownian particle out of a potential well: microwave-enhanced escape from the zero-voltage state of a Josephson junction. Phys. Rev. B 36(1), 58–73 (1987)CrossRefGoogle Scholar
  4. 4.
    Vijay, R., Devoret, M., Siddiqi, I.: Invited review article: the Josephson bifurcation amplifier. Rev. Sci. Instrum. 80(11), 111101 (2009)CrossRefGoogle Scholar
  5. 5.
    Aldridge, J., Cleland, A.: Noise-enabled precision measurements of a Duffing nanomechanical resonator. Phys. Rev. Lett. 94(15), 156403 (2005)CrossRefGoogle Scholar
  6. 6.
    Stambaugh, C., Chan, H.: Noise-activated switching in a driven nonlinear micromechanical oscillator. Phys. Rev. B 73(17), 172302 (2006)CrossRefGoogle Scholar
  7. 7.
    Harne, R.L., Wang, K.W.: Harnessing Bistable Structural Dynamics: For Vibration Control, Energy Harvesting and Sensing. Wiley, Hoboken (2017)CrossRefzbMATHGoogle Scholar
  8. 8.
    Johnson, D.R., Thota, M., Semperlotti, F., Wang, K.W.: On achieving high and adaptable damping via a bistable oscillator. Smart Mater. Struct. 22(11), 115027 (2013)CrossRefGoogle Scholar
  9. 9.
    Wu, Z., Harne, R.L., Wang, K.W.: Energy harvester synthesis via coupled linear-bistable system with multi-stable dynamics. ASME J. Appl. Mech. 81(6), 061005 (2014)CrossRefGoogle Scholar
  10. 10.
    Harne, R.L., Wang, K.W.: A bifurcation-based coupled linear-bistable system for microscale mass sensing. J. Sound Vib. 333(8), 2241–2252 (2014)CrossRefGoogle Scholar
  11. 11.
    Koper, M.: Non-linear phenomena in electrochemical systems. J. Chem. Soc. Faraday Trans. 94(10), 1369–1378 (1998)CrossRefGoogle Scholar
  12. 12.
    Dakos, V., van Nes, E., D’Odorico, P., Scheffer, M.: Robustness of variance and autocorrelation as indicators of critical slowing down. Ecology 93(2), 264–271 (2012)CrossRefGoogle Scholar
  13. 13.
    Scheffer, M., Carpenter, S., Lenton, T., Bascompte, J., Brock, W., Dakos, V., van de Koppel, J., van de Leemput, I., Levin, S., van Nes, E., Pascual, M., Vandermeer, J.: Anticipating critical transitions. Science 338(6105), 344–348 (2012)CrossRefGoogle Scholar
  14. 14.
    D’Souza, K., Epureanu, B., Pascual, M.: Forecasting bifurcations from large perturbation recoveries in feedback ecosystems. PLoS ONE 10(9), e0137779 (2015)CrossRefGoogle Scholar
  15. 15.
    Lenton, T.: Early warning of climate tipping points. Nat. Clim. Change 1(4), 201–209 (2011)CrossRefGoogle Scholar
  16. 16.
    Scheffer, M., Bascompte, J., Brock, W., Brovkin, V., Carpenter, S., Dakos, V., Held, H., van Nes, E., Rietkerk, M., Sugihara, G.: Early-warning signals for critical transitions. Nature 461(7260), 53–59 (2009)CrossRefGoogle Scholar
  17. 17.
    Hanggi, P.: Escape from a metastable state. J. Stat. Phys. 42(1–2), 105–148 (1986)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Meunier, C., Verga, A.: Noise and bifurcations. J. Stat. Phys. 50(1–2), 345–375 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Dykman, M., Schwartz, I., Shapiro, M.: Scaling in activated escape of underdamped systems. Phys. Rev. E 72(2), 021102 (2005)CrossRefGoogle Scholar
  20. 20.
    Lu, C.-H., Evan-Iwanowski, R.: The nonstationary effects on a softening Duffing oscillator. Mech. Res. Commun. 21(6), 555–564 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mandel, P., Erneux, T.: The slow passage through a steady bifurcation: delay and memory effects. J. Stat. Phys. 48(5–6), 1059–1070 (1987)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Breban, R., Nusse, H., Ott, E.: Scaling properties of saddle-node bifurcations on fractal basin boundaries. Phys. Rev. E 68(6), 066213 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Berglund, N., Gentz, B.: Noise-Induced Phenomena in Slow-Fast Dynamical Systems: A Sample-Paths Approach. Springer, New York (2006)zbMATHGoogle Scholar
  24. 24.
    Kuehn, C.: Multiple Time Scale Dynamics. Springer, New York (2015)CrossRefzbMATHGoogle Scholar
  25. 25.
    Nicolis, C., Nicolis, G.: Dynamical responses to time-dependent control parameters in the presence of noise: a normal form approach. Phys. Rev. E 89(2), 022903 (2014)CrossRefGoogle Scholar
  26. 26.
    Miller, N., Shaw, S.: Escape statistics for parameter sweeps through bifurcations. Phys. Rev. E 85(4), 046202 (2012)CrossRefGoogle Scholar
  27. 27.
    Kim, J., Harne, R.L., Wang, K.W.: Predicting non-stationary and stochastic activation of saddle-node bifurcation. J. Comput. Nonlinear Dyn. 12(1), 011009 (2016)CrossRefGoogle Scholar
  28. 28.
    Leine, R., Van Campen, D., Keultjes, W.: Stick-slip whirl interaction in drillstring dynamics. J. Vib. Acoust. 124(2), 209–220 (2002)CrossRefGoogle Scholar
  29. 29.
    Sung, C., Yu, W.: Dynamics of a harmonically excited impact damper: bifurcations and chaotic motion. J. Sound Vib. 158(2), 317–329 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Holmes, P., Full, R., Koditschek, D., Guckenheimer, J.: The dynamics of legged locomotion: models, analyses, and challenges. SIAM Rev. 48(2), 207–304 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Iqbal, S., Zang, X., Zhu, Y., Zhao, J.: Bifurcations and chaos in passive dynamic walking: a review. Robot. Auton. Syst. 62(6), 889–909 (2014)CrossRefGoogle Scholar
  32. 32.
    Chua, L.: Chua’s circuit 10 years later. Int. J. Circuit Theory Appl. 22(4), 279–305 (1994)CrossRefGoogle Scholar
  33. 33.
    di Bernardo, M., Budd, C., Champneys, A., Kowalczyk, P., Nordmark, A., Tost, G., Piiroinen, P.: Bifurcations in nonsmooth dynamical systems. SIAM Rev. 50(4), 629–701 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Harne, R.L., Wang, K.W.: Robust sensing methodology for detecting change with bistable circuitry dynamics tailoring. Appl. Phys. Lett. 102(20), 203506 (2013)CrossRefGoogle Scholar
  35. 35.
    Kim, J., Harne, R.L., Wang, K.W.: Enhancing structural damage identification robustness to noise and damping with integrated bistable and adaptive piezoelectric circuitry. J. Vib. Acoust. 137(1), 011003 (2015)CrossRefGoogle Scholar
  36. 36.
    Zhao, X., Schaeffer, D., Berger, C., Krassowska, W., Gauthier, D.: Cardiac alternans arising from an unfolded border-collision bifurcation. J. Comput. Nonlinear Dyn. 3(4), 041004 (2008)CrossRefGoogle Scholar
  37. 37.
    Caballé, J., Jarque, X., Michetti, E.: Chaotic dynamics in credit constrained emerging economies. J. Econ. Dyn. Control 30(8), 1261–1275 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Mosekilde, E., Laugesen, J.: Nonlinear dynamic phenomena in the Beer model. Syst. Dyn. Rev. 23(2–3), 229–252 (2007)CrossRefGoogle Scholar
  39. 39.
    Simpson, D.: Bifurcations in Piecewise-Smooth Continuous Systems. World Scientific, Singapore (2010)CrossRefzbMATHGoogle Scholar
  40. 40.
    Leine, R., Nijmeijer, H.: Dynamics and Bifurcations of Non-Smooth Mechanical Systems. Springer, New York (2013)zbMATHGoogle Scholar
  41. 41.
    di Bernardo, M., Budd, C., Champneys, A., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications. Springer, New York (2008)zbMATHGoogle Scholar
  42. 42.
    Masri, S., Caughey, T.: On the stability of the impact damper. J. Appl. Mech. 33(3), 586–592 (1966)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Natsiavas, S.: Periodic response and stability of oscillators with symmetric trilinear restoring force. J. Sound Vib. 134(2), 315–331 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Shaw, S., Holmes, P.: A periodically forced piecewise linear oscillator. J. Sound Vib. 90(1), 129–155 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Luo, A.: The mapping dynamics of periodic motions for a three-piecewise linear system under a periodic excitation. J. Sound Vib. 283(3), 723–748 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Kuehn, C.: Scaling of saddle-node bifurcations: degeneracies and rapid quantitative changes. J. Phys. A Math. Theor. 42(4), 045101 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Dankowicz, H., Nordmark, A.B.: On the origin and bifurcations of stick-slip oscillations. Phys. D 136(3), 280–302 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Risken, H.: The Fokker–Planck Equation. Springer, Berlin (1989)CrossRefzbMATHGoogle Scholar
  49. 49.
    Tamaševičius, A., Mykolaitis, G., Pyragas, V., Pyragas, K.: Delayed feedback control of periodic orbits without torsion in nonautonomous chaotic systems: theory and experiment. Phys. Rev. E 76(2), 026203 (2007)CrossRefGoogle Scholar
  50. 50.
    Forgoston, E., Schwartz, I.: Escape rates in a stochastic environment with multiple scales. SIAM J. Appl. Dyn. Syst. 8(3), 1190–1217 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Holmes, P.: A nonlinear oscillator with a strange attractor. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 292(1394), 419–448 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Kovacic, I., Brennan, M.J.: The Duffing Equation: Nonlinear Oscillators and Their Behaviour. Wiley, Hoboken (2011)CrossRefzbMATHGoogle Scholar
  53. 53.
    Rizzoni, G.: Fundamentals of Electrical Engineering. McGraw-Hill, New York (2009)Google Scholar
  54. 54.
    Higham, D.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525–546 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of MichiganAnn ArborUSA

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