Abstract
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.
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References
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)
Arnold, L.: Random Dynamical Systems. Springer, New York (1998)
Arnold, L., Crauel, H.: Random dynamical systems. Lect. Notes Mat. 1486, 1–22 (1991)
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)
Baxendale, P.H.: A stochastic hopf bifurcation. Probab. Theory Rel. Fields 99, 581–616 (1994)
Davies, H.G.: Random vibrations of a beam impacting stops. J. Sound Vib. 68(4), 479–487 (1980)
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)
Dimentberg, M.: Statistical Dynamics of Nonlinear and Time-Varying Systems. Research Studies Press, Taunton (1988)
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)
Dimentberg, M.F., Iourtchenko, D.V.: Random vibrations with impacts: a review. Nonlinear Dyn. 36(2–4), 229–254 (2004)
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)
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)
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)
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)
Filippov, A.F.: Differential equations with discontinuous righthand sides. Kluwer Academic Publishers, Berlin (1988)
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), 223–240
Ibrahim, R.: Recent advances in vibro-impact dynamics and collision of ocean vessels. J. Sound Vib. 333, 5900–5916 (2014)
Ibrahim, R.A.: Vibro-Impact Dynamics Modeling. Mapping and Applications. Springer, New York (2009)
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)
Ivanov, A.P.: Impact oscillations: linear theory of stability and bifurcations. J. Sound Vib. 178(3), 361–378 (1994)
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)
Kim, S., Park, S.H., Ryn, C.: Noise-enhanced multistabilty in coupled oscillator systems. Phys. Rev. Lett. 78, 1616–1619 (1997)
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)
Kumar, P., Gupta, S.: Investigations on the bifurcation of a noisy Duffing–van der Pol oscillator. Probab. Eng. Mech. (accepted)
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)
Narayanan, S., Jayaraman, K.: Chaotic vibration in a non-linear oscillator with coulomb damping. J. Sound Vib. 146(1), 17–31 (1991)
Nordmark, A.B.: Non-periodic motion cause by grazing incidence in an impact oscillator. J. Sound Vib. 145, 279–297 (1991)
Phillis, Y.A.: Entropy stability of continuous dynamic system. Int. J. Control 35, 323–340 (1982)
Pilipchuk, V.: Non-smooth spatio-temporal coordinates in nonlinear dynamics. http://arxiv.org/pdf/1101.4597v1.pdf (2013)
Ramasubramanian, K., Sriram, M.S.: A comparative study of computation of lyapunov spectra with different algorithms. Physica D 139, 72–86 (2000)
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)
Schenk-Hoppe, K.R.: Bifurcation scenario of the noisy Duffing stochastic-Van der Pol oscillator. Nonlinear Dyn. 11, 255–274 (1996)
Namachchivaya, N.S., Park, J.: Stochastic dynamics of impact oscillators. ASME J. Appl. Mech. 72(6), 862–870 (2005)
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)
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)
Wei, J., Leng, G.: Lyapunov exponent and chaos of Duffing’s equation perturbed by white noise. Appl. Math. Comput. 88, 77–93 (1997)
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)
Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining lyapunov exponents from a time series. Physica D 16, 285–317 (1985)
Zhu, H.T.: Stochastic response of vibro-impact Duffing oscillator under eternal and parametric gaussian white noises. J. Sound Vib. 333, 954–961 (2014)
Zhuravlev, V.F.: A method for analyzing vibro-impact systems by means of special functions. Mech. Solids 11, 23–27 (1976)
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Kumar, P., Narayanan, S. & Gupta, S. Stochastic bifurcations in a vibro-impact Duffing–Van der Pol oscillator. Nonlinear Dyn 85, 439–452 (2016). https://doi.org/10.1007/s11071-016-2697-1
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DOI: https://doi.org/10.1007/s11071-016-2697-1