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Chaos and bifurcation analysis of stochastically excited discontinuous nonlinear oscillators

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Abstract

Dynamics of discontinuous nonlinear systems subjected to random excitation is studied. Such systems occur in many mechanical and aerospace applications involving impact, friction, clearance, backlash, freeplay etc. These systems are characterized by sharp switches in dynamical behaviour described by discontinuous stochastic differential equations. An adaptive time step** approach is developed in combination with a bisection algorithm to locate precisely the discontinuity point in the numerical integration advanced by the Milstein method. The Brownian tree approach is used to direct the integration along the correct Brownian path. The examples of a Duffing oscillator with one- and two-sided impacts and a linear oscillator with a nonlinear discontinuous dry friction-type damper (Coulomb dam**) subjected to combined harmonic and white noise excitations are considered. Stable periodic motion, D-bifurcation and chaotic dynamics are exhibited in different parametric regimes. The path-wise numerical integration procedure demonstrates the accuracy and efficiency of the proposed scheme in the dynamic analysis of the noisy vibro-impact oscillator and the friction oscillator.

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References

  1. Arnold, L., Sri Namachchivaya, N., Schenk-Hoppe, K.R.: Toward an understanding of stochastic Hopf bifurcation: A case study. Int. J. Bifurc. Chaos 6, 1947–1975 (1996)

    MathSciNet  MATH  Google Scholar 

  2. Aziz, M., Vakakis, A., Manevich, L.: Exact solutions of the problem of the vibro-impact oscillations of a discrete system with two degrees of freedom. J. Appl. Math. Mech. 63, 527–530 (1999)

    MathSciNet  MATH  Google Scholar 

  3. Baxendale, P.H.: A stochastic Hopf bifurcation. Probab. Th. Rel. Fields 99, 581–616 (1994)

    MathSciNet  MATH  Google Scholar 

  4. Brogliato, B.: Nonsmooth Mechanics : Models, Dynamics and Control. Springer, Berlin (2016)

    MATH  Google Scholar 

  5. Brogliato, B., Niculescu, S.I., Orhant, P.: On the control of finite-dimensional mechanical systems with unilateral constants. IEEE Trans. Autom. Control 146, 200–215 (1997)

    MATH  Google Scholar 

  6. Burrage, P.M., Burrage, K.: A variable stepsize implementation for stochastic differential equation. SIAM J. Sci. Comput. 24(3), 848–864 (2003)

    MathSciNet  MATH  Google Scholar 

  7. Cheng, J., Xu, H.: Periodic motions, bifurcation, and hysteresis of the vibro-impact system. Mech. Based Des. Struct. Mach. 35(2), 179–203 (2007)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. Dimentberg, M.F., Gaidai, O., Naess, A.: Vibro-impact dynamics of ocean systems and related problems. In: Ibrahim, R., Babitsky, V., Okuma, M. (eds.) Lecture Notes in Applied and Computational Mechanics, vol. 44, pp. 67–78 (2009)

  10. Dimentberg, M.F., Iourtchenko, D.V.: Random vibrations with impacts: A review. Nonlinear Dyn. 36, 229–254 (2004)

    MathSciNet  MATH  Google Scholar 

  11. Dimentberg, M.F., Menyailov, A.: Response of a single-mass vibro-impact system to white noise random excitation. ZAMM-J. Appl. Math. Mech. 59(12), 709–716 (1979)

    MATH  Google Scholar 

  12. Dimentberg, M.F., Gaidai, O., Naess, A.: Random vibrations with strongly inelastic impacts: Response PDF by the path integration method. Int. J. Non-Linear Mech. 44, 791–796 (2009)

    MATH  Google Scholar 

  13. 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)

    MathSciNet  MATH  Google Scholar 

  14. Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers, (1988)

  15. Gaines, J., Lyons, T.: Variable step size control in the numerical solution of stochastic differential equations. SIAM J. Appl. Math. 57, 1455–1484 (1997)

    MathSciNet  MATH  Google Scholar 

  16. Glocker, C.: Set-Valued Force Laws, Dynamics of Non-Smooth Systems. Springer-Verlag, Berlin (2001)

    MATH  Google Scholar 

  17. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, Berlin (1983)

    MATH  Google Scholar 

  18. Hoskins, R.F.: Delta Functions : Introduction to Generalised Functions. Woodhead Publishing, India (2011)

    MATH  Google Scholar 

  19. Ibrahim, R.A.: Vibro-Impact Dynamics Modeling, Map** and Applications. Springer, Berlin (2009)

    MATH  Google Scholar 

  20. Ibrahim, R.A.: Rceent advances in vibro-impact dynamics and collision of ocean vessels. J. Sound Vib. 333, 5900–5916 (2014)

    Google Scholar 

  21. Iourtchenko, D.V., Eirik, M., Naess, A.: Response probability density functions of strongly non-linear systems by the path integration method. Int. J. Non-Linear Mech. 41, 693–705 (2006)

    MathSciNet  MATH  Google Scholar 

  22. Ivanov, A.P.: Impact oscillations: Linear theory of stability and bifurcations. J. Sound Vib. 178, 361–378 (1994)

    MathSciNet  MATH  Google Scholar 

  23. **, 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)

    MathSciNet  MATH  Google Scholar 

  24. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equation. Springer-Verlag, Berlin (1992)

    MATH  Google Scholar 

  25. Kumar, P., Narayanan, S., Gupta, S.: Numerical simulation of stochastically excited discontinuous nonlinear systems through adaptive time step**. In: 10th European Nonlinear Dynamics Conference, p. Accepted (2020)

  26. 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)

    Google Scholar 

  27. Kumar, P., Narayanan, S., Gupta, S.: Investigations on the bifurcation of a noisy Duffing-Van der Pol oscillator. Probab. Eng. Mech. 45, 70–86 (2016)

    Google Scholar 

  28. Kumar, P., Narayanan, S., Gupta, S.: Stochastic bifurcations in a vibro-impact Duffing-Van der Pol oscillator. Nonlinear Dyn. 85, 439–52 (2016)

    MathSciNet  Google Scholar 

  29. Kumar, P., Narayanan, S., Gupta, S.: Bifurcation analysis of a stochastically excited vibro-impact Duffing-Van der Pol oscillator with bilateral rigid barriers. Int. J. Mech. Sci. 127, 103–117 (2017)

    Google Scholar 

  30. Lamba, H.: An adaptive time-step** algorithm for stochastic differential equations. J. Comput. Appl. Math. 161(2), 417–430 (2003)

    MathSciNet  MATH  Google Scholar 

  31. Lenci, S., Rega, G.: Global chaos control in a periodically forced oscillator. In: Bajaj, A.K., Namachchivaya, N.S., Francheck, M.A. (Eds.), Proc. ASME Int. Mech. Engg. Congress on Nonlinear Dynamics and Control, Atlanta, Georgia, vol. DE–91, pp. 111–116 (1996)

  32. Lenci, S., Rega, G.: Procedure for reducing the chaotic response region in an impact mechanical system. Nonlinear Dyn. 15, 391–409 (1998)

    MathSciNet  MATH  Google Scholar 

  33. Levy, P.: Monographie des Probabilites Gauthier–Villars. Paris, (1948)

  34. Müller, P.: Calculation of Lyapunov exponent for dynamics system with discontinuities. Chaos Solitons and Fractals 5, 1671–1681 (1995)

    MathSciNet  MATH  Google Scholar 

  35. Narayanan, S., Jayaraman, K.: Chaotic vibration in a non-linear oscillator with coulomb dam**. J. Sound Vib. 146(1), 17–31 (1991)

    Google Scholar 

  36. Nordmark, A.B.: Non-periodic motion cause by grazing incidence in an impact oscillator. J. Sound Vib. 145, 279–297 (1991)

    Google Scholar 

  37. Nordmark, A., Dankowicz, H., Champneys, A.: Discontinuity-induced bifurcations in systems with impacts and friction: Discontinuities in the impact law. Int. J. Non-Linear Mech. 44, 1011–1023 (2009)

    MATH  Google Scholar 

  38. Oseledec, V.I.: A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19, 197–231 (1968)

    MathSciNet  Google Scholar 

  39. Oza, H.B., Orlov, Y.V., Spurgeon, S.K.: Finite time stabilization of a perturbed double integrator with unilateral constraints. Math. Comput. Simul. 95, 200–212 (2014)

    MathSciNet  Google Scholar 

  40. Piiroinen, P., Kuznetsov, Y.: An event driven method to simulate Filippov systems with accurate computing of sliding motions. ACM Trans. Math. Softw. 34, 1–24 (2008)

    MathSciNet  MATH  Google Scholar 

  41. Pilipchuk, V.N.: Non-smooth spatial and temporal substitutions in impact dynamics. In: Andrianov I., Manevich A., Mikhlin Y., Gendelman O. (eds) Problems of Nonlinear Mechanics and Physics of Materials. Advanced Structured Material vol. 94, pp. 119 – 140 (2019)

  42. Pilipchuk, V.N.: Non-smooth spatio-temporal coordinates in nonlinear dynamics. pp. 1 – 36 (2013) ar**v: 1101.4597v1

  43. Pilipchuk, V.N.: Temporal transformations and visualization diagrams for nonsmooth periodic motions. Int. J. Bifurc. Chaos 15(06), 1879–1899 (2005)

    MathSciNet  MATH  Google Scholar 

  44. Pilipchuk, V.N.: Nonlinear Dynamics: Between Linear and Impact Limits. Springer, Berlin, Heidelberg (2010)

    MATH  Google Scholar 

  45. Pilipchuk, V.N.: Closed-form solutions for oscillators with inelastic impacts. J. Sound Vib. 359, 154–167 (2015)

    Google Scholar 

  46. Pilipchuk, V.N., Ibrahim, R.A.: Dynamics of a two-pendulum model with impact interaction and an elastic support. Nonlinear Dyn. 21(03), 221–247 (2000)

    MathSciNet  MATH  Google Scholar 

  47. Popp, K.: Non-smooth mechanical systems. J. Appl. Math. Mech. 64(5), 765–772 (2000)

    MATH  Google Scholar 

  48. P\(\mathring{\rm u}\)st, L., Peterka, F.: Impact oscillator with Hertz’s model of contact. Meccanica 38, 99–116 (2003)

  49. Rega, G., Lenci, S.: Nonsmooth dynamics, bifurcation and control in an impact system. Syst. Anal. Modell. Simul. 43, 343–360 (2003)

    MathSciNet  MATH  Google Scholar 

  50. Richtmyer, R.D.: Principles of Advanced Mathematical Physics. Springe, Berlinr (1985)

    MATH  Google Scholar 

  51. Risken, H.: The Fokker- Planck Equation : Methods of Solution and Applications. Springer-Verlag, New York (1989)

    MATH  Google Scholar 

  52. Santhosh, B., Padmanabhan, C., Narayanan, S.: Numeric-analytic solutions of the smooth and discontinuous oscillator. Int. J. Mech. Sci. 84, 102–119 (2014)

    Google Scholar 

  53. Schwartz, L.: Théorie des Distributions. Hermann, Paris (1966)

    MATH  Google Scholar 

  54. Shampine, L.F., Thompson, S.: Event location for ordinary differential equations. Comput. Math. Appl. 39, 43–54 (2000)

    MathSciNet  MATH  Google Scholar 

  55. Sotiropoulos, V., Kaznessis, Y.N.: An adaptive time step scheme for a system of stochastic differential equations with multiple multiplicative noise: Chemical Langevin equation, a proof of concept. J. Chem. Phys. 128(1), 014103 (2008)

    Google Scholar 

  56. Wedig, W.: Dynamic stability of beams under axial forces-Lyapunov exponents for general fluctuating loads. In: W. Kr-ig (ed.) Proceedings Eurodyn’90,Conference on Structural Dynamics, vol. 1, pp. 57–64 (1990)

  57. Wei, S.T., Pierre, C.: Effects of dry friction dam** on the occurrence of localized forced vibration in nearly cyclic structures. J. Sound Vib. 129, 397–416 (1989)

    Google Scholar 

  58. Wiercigroch, M.: Modeling of dynamical systems with motion dependent discontinuities. Chaos, Solitons and Fractals 11, 2429–42 (2000)

    MathSciNet  MATH  Google Scholar 

  59. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Phys. D 16, 285–317 (1985)

    MathSciNet  MATH  Google Scholar 

  60. Wong, E., Zakai, M.: On the relation between ordinary and stochastic differential equation. Int. J. Eng. Sci. 3, 213–229 (1965)

    MathSciNet  MATH  Google Scholar 

  61. Zhuravlev, V.F.: A method for analyzing vibro-impact systems by means of special functions. Mech. Solids 11, 23–27 (1976)

    MathSciNet  Google Scholar 

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  1. Dynamic Analysis Group, Bharat Heavy Electricals Limited, Nagpur, 440001, India

    Pankaj Kumar

  2. Indian Institute of Information Technology (Design and Manufacturing), Kancheepuram, Chennai, 600 127, India

    S. Narayanan

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  1. Pankaj Kumar
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Kumar, P., Narayanan, S. Chaos and bifurcation analysis of stochastically excited discontinuous nonlinear oscillators. Nonlinear Dyn 102, 927–950 (2020). https://doi.org/10.1007/s11071-020-05960-5

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