Nonlinear Dynamics

, Volume 85, Issue 1, pp 439–452 | Cite as

Stochastic bifurcations in a vibro-impact Duffing–Van der Pol oscillator

  • Pankaj Kumar
  • S. Narayanan
  • Sayan Gupta
Original Paper


The stochastic bifurcations in a vibro-impact Duffing–Van der Pol oscillator, subjected to white noise excitations, are investigated. Bifurcations in noisy systems occur either due to topological changes in the phase space—known as D-bifurcations—or due to topological changes associated with the stochastic attractors—known as P-bifurcations. In either case, the singularities in the phase space near the grazing orbits due to impact lead to inherent difficulties in bifurcation analysis. Loss of dynamic stability—or D-bifurcations—is analyzed through computation of the largest Lyapunov exponent using the Nordmark–Poincare mapping that enables bypassing the problems associated with discontinuities. For P-bifurcation analysis, the steady-state solution of the Fokker–Planck equation is computed after applying suitable non-smooth coordinate transformations and mapping the problem into a continuous domain. A quantitative measure for P-bifurcations has been carried out using a newly developed measure based on Shannon entropy. A comparison of the stability domains obtained from P-bifurcation and D-bifurcation analyses is presented which reveals that these bifurcations need not occur in same regimes.


Vibro-impact Duffing–Van der Pol oscillator Zhuravlev–Ivanov transformation Nordmark–Poincare mapping Shannon entropy Stochastic bifurcation 


  1. 1.
    Arecchi, F., Badii, R., Politi, A.: Generalized multistability and noise-induced jumps in a nonlinear dynamical system. Phys. Rev. A 32(1), 402–408 (1985)CrossRefGoogle Scholar
  2. 2.
    Arnold, L.: Random Dynamical Systems. Springer, New York (1998)CrossRefMATHGoogle Scholar
  3. 3.
    Arnold, L., Crauel, H.: Random dynamical systems. Lect. Notes Mat. 1486, 1–22 (1991)MathSciNetGoogle Scholar
  4. 4.
    Arnold, L.: Sri Namachchivaya, N., Schenk-Hoppe, K.R.: Toward an understanding of stochastic hopf bifurcation: a case study. Int. J. Bifurc Chaos 6(11), 1947–1975 (1996)CrossRefGoogle Scholar
  5. 5.
    Baxendale, P.H.: A stochastic hopf bifurcation. Probab. Theory Rel. Fields 99, 581–616 (1994)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Davies, H.G.: Random vibrations of a beam impacting stops. J. Sound Vib. 68(4), 479–487 (1980)CrossRefMATHGoogle Scholar
  7. 7.
    Di Bernardo, M., Nordmark, A., Olivar, G.: Discontinuity-induced bifurcations of equilibria in piecewise smooth and impacting dynamical systems. Physica D 237, 119–136 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dimentberg, M.: Statistical Dynamics of Nonlinear and Time-Varying Systems. Research Studies Press, Taunton (1988)MATHGoogle Scholar
  9. 9.
    Dimentberg, M., Naess, A., Gaidai, O.: Random vibrations with strongly inelastic impacts: response pdf by pi method. Int. J. Nonlinear Mech. 44, 791–796 (2009)CrossRefMATHGoogle Scholar
  10. 10.
    Dimentberg, M.F., Iourtchenko, D.V.: Random vibrations with impacts: a review. Nonlinear Dyn. 36(2–4), 229–254 (2004)Google Scholar
  11. 11.
    Dimentberg, M.F., Menyailov, A.: Response of a single-mass vibroimpact system to white noise random excitation. ZAMM J. Appl. Math. Mech. 59(12), 709–716 (1979)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Feng, J., Xu, W.: Analysis of bifurcation for nonlinear stochastic non-smooth vibro impact systems via top Lyapunov exponent. Appl. Math. Comput. 213, 577–586 (2009)MathSciNetMATHGoogle Scholar
  13. 13.
    Feng, J., Xu, W., Wang, R.: Stochastic response of vibro impact Duffing oscillator excited by additive gaussian noise. J. Sound Vib. 309(3–5), 730–738 (2008)Google Scholar
  14. 14.
    Feng, J.Q., Xu, W., Rong, H.W., Wang, R.: Stochastic response of Duffing?van der pol vibro impact system under additive and multiplicative random excitation. Int. J. Non-Linear Mech. 44(1), 51–57 (2009)CrossRefMATHGoogle Scholar
  15. 15.
    Filippov, A.F.: Differential equations with discontinuous righthand sides. Kluwer Academic Publishers, Berlin (1988)CrossRefGoogle Scholar
  16. 16.
    Huang, Z.L., Liu, Z.H., Zhu, W.: Stationary response of multi degree-of-freedom vibro-impact system under white noise excitations. J. Sound Vib. 275(1–2), 223–240Google Scholar
  17. 17.
    Ibrahim, R.: Recent advances in vibro-impact dynamics and collision of ocean vessels. J. Sound Vib. 333, 5900–5916 (2014)CrossRefGoogle Scholar
  18. 18.
    Ibrahim, R.A.: Vibro-Impact Dynamics Modeling. Mapping and Applications. Springer, New York (2009)CrossRefGoogle Scholar
  19. 19.
    Iourtchenko, D.V., Song, L.L.: Numerical investigation of a response probability density function of stochastic vibroimpact systems with inelastic inputs. Int. J. Non-Linear Mech. 41(3), 447–455 (2006)CrossRefMATHGoogle Scholar
  20. 20.
    Ivanov, A.P.: Impact oscillations: linear theory of stability and bifurcations. J. Sound Vib. 178(3), 361–378 (1994)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Jin, L., Lu, Q., Twizell, E.H.: A method for calculating the spectrum of lyapunov exponents by local maps in non-smooth impact vibrating systems. J. Sound Vib. 298, 1019–1033 (2006)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Kim, S., Park, S.H., Ryn, C.: Noise-enhanced multistabilty in coupled oscillator systems. Phys. Rev. Lett. 78, 1616–1619 (1997)CrossRefGoogle Scholar
  23. 23.
    Kumar, P., Narayanan, S., Gupta, S.: Finite element solution of Fokker?Planck equation of nonlinear oscillators subjected to colored non-gaussian noise. Probab. Eng. Mech. 38, 143–155 (2014)CrossRefGoogle Scholar
  24. 24.
    Kumar, P., Gupta, S.: Investigations on the bifurcation of a noisy Duffing–van der Pol oscillator. Probab. Eng. Mech. (accepted)Google Scholar
  25. 25.
    Luo, G.W., Chu, Y.L., Zang, Y.L., Zang, J.G.: Double Neimark?Sacker bifurcation and torus bifurcation of a class of vibratory systems with symmetrical rigid stops. J. Sound Vib. 298(4), 154–179 (2006)Google Scholar
  26. 26.
    Narayanan, S., Jayaraman, K.: Chaotic vibration in a non-linear oscillator with coulomb damping. J. Sound Vib. 146(1), 17–31 (1991)CrossRefGoogle Scholar
  27. 27.
    Nordmark, A.B.: Non-periodic motion cause by grazing incidence in an impact oscillator. J. Sound Vib. 145, 279–297 (1991)CrossRefGoogle Scholar
  28. 28.
    Phillis, Y.A.: Entropy stability of continuous dynamic system. Int. J. Control 35, 323–340 (1982)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Pilipchuk, V.: Non-smooth spatio-temporal coordinates in nonlinear dynamics. (2013)
  30. 30.
    Ramasubramanian, K., Sriram, M.S.: A comparative study of computation of lyapunov spectra with different algorithms. Physica D 139, 72–86 (2000)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Rong, H., Wang, X., Xu, W., Feng, T.: Resonant response of a non-linear vibro impact system to combined deterministic harmonic and random excitations. Int. J. Non-Linear Mech. 45, 474–481 (2010)CrossRefGoogle Scholar
  32. 32.
    Schenk-Hoppe, K.R.: Bifurcation scenario of the noisy Duffing stochastic-Van der Pol oscillator. Nonlinear Dyn. 11, 255–274 (1996)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Namachchivaya, N.S., Park, J.: Stochastic dynamics of impact oscillators. ASME J. Appl. Mech. 72(6), 862–870 (2005)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Wagg, D., Bishop, S.R.: Chatter, sticking and chaotic impacting motion in a two degree of freedom impact oscillator. Int. J. Bifurc. Chaos 11(1), 57–71 (2001)CrossRefGoogle Scholar
  35. 35.
    Wedig, W.: Dynamic stability of beams under axial forces-lyapunov exponents for general fluctuating loads. In: Kr-ig, W. (ed.) Proceedings Eurodyn’90, Conference on Structural Dynamics, vol. 1, pp. 57–64 (1990)Google Scholar
  36. 36.
    Wei, J., Leng, G.: Lyapunov exponent and chaos of Duffing’s equation perturbed by white noise. Appl. Math. Comput. 88, 77–93 (1997)MathSciNetMATHGoogle Scholar
  37. 37.
    Wei, S.T., Pierre, C.: Effects of dry friction damping on the occurrence of localized forced vibration in nearly cyclic structures. J. Sound Vib. 129, 397–416 (1989)CrossRefGoogle Scholar
  38. 38.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining lyapunov exponents from a time series. Physica D 16, 285–317 (1985)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Zhu, H.T.: Stochastic response of vibro-impact Duffing oscillator under eternal and parametric gaussian white noises. J. Sound Vib. 333, 954–961 (2014)CrossRefGoogle Scholar
  40. 40.
    Zhuravlev, V.F.: A method for analyzing vibro-impact systems by means of special functions. Mech. Solids 11, 23–27 (1976)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringIndian Institute of Technology MadrasChennaiIndia
  2. 2.Indian Institute of Information Technology (Design and Manufacturing) KancheepuramChennaiIndia
  3. 3.Department of Applied MechanicsIndian Institute of Technology MadrasChennaiIndia

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