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Noise-induced multistability in the square root map

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Abstract

The effects of small-amplitude additive Gaussian white noise on the one-dimensional square root map are investigated. In particular, the focus is on the unexpected effects noise of varying amplitudes has on the system for parameter regions just outside intervals of multistability. It is shown that in these regions periodic behaviour that is unstable in the deterministic system can be effectively stabilised by the addition of noise of an appropriate amplitude. Features of noise-induced transitions from stable to stabilised unstable periodic behaviour are highlighted, and it is shown how these features can be understood by examining relative levels of expansion and contraction in the deterministic map.

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References

  1. Arnold, L., Boxler, P.: Stochastic bifurcation: instructive examples in dimension one. Diffusion Processes and Related Problems in Analysis, vol. II, pp. 241–255. Springer, Berlin (1992)

    MATH  Google Scholar 

  2. Avrutin, V., Dutta, P.S., Schanz, M., Banerjee, S.: Influence of a square-root singularity on the behaviour of piecewise smooth maps. Nonlinearity 23(2), 445 (2010)

    Article  MathSciNet  Google Scholar 

  3. Bernardo, M., Budd, C., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications, vol. 163. Springer, Berlin (2008)

    MATH  Google Scholar 

  4. Bishop, S.: Impact oscillators. Philos. Trans. R. Soc. Lond. A: Math. Phys. Eng. Sci. 347(1683), 347–351 (1994)

    Article  Google Scholar 

  5. Chin, W., Ott, E., Nusse, H.E., Grebogi, C.: Grazing bifurcations in impact oscillators. Phys. Rev. E 50(6), 4427 (1994)

    Article  MathSciNet  Google Scholar 

  6. Di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P., Nordmark, A.B., Tost, G.O., Piiroinen, P.T.: Bifurcations in nonsmooth dynamical systems. SIAM Rev. 50(4), 629–701 (2008)

    Article  MathSciNet  Google Scholar 

  7. Guckenheimer, J.: On the bifurcation of maps of the interval. Invent. Math. 39(2), 165–178 (1977)

    Article  MathSciNet  Google Scholar 

  8. Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42. Springer, Berlin (2013)

    MATH  Google Scholar 

  9. Kraut, S., Feudel, U., Grebogi, C.: Preference of attractors in noisy multistable systems. Phys. Rev. E 59(5), 5253 (1999)

    Article  Google Scholar 

  10. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, vol. 112. Springer, Berlin (2013)

    Google Scholar 

  11. Linz, S., Lücke, M.: Effect of additive and multiplicative noise on the first bifurcations of the logistic model. Phys. Rev. A 33(4), 2694 (1986)

    Article  Google Scholar 

  12. Longtin, A.: Effects of noise on nonlinear dynamics. Nonlinear Dynamics in Physiology and Medicine, pp. 149–189. Springer, Berlin (2003)

    Chapter  Google Scholar 

  13. Medeiros, E.S., Caldas, I.L., Baptista, M.S., Feudel, U.: Trapping phenomenon attenuates the consequences of tipping points for limit cycles. Sci. Rep. 7, 42,351 (2017)

    Article  Google Scholar 

  14. Nordmark, A.B.: Non-periodic motion caused by grazing incidence in an impact oscillator. J. Sound Vib. 145(2), 279–297 (1991)

    Article  Google Scholar 

  15. Nordmark, A.B.: Universal limit mapping in grazing bifurcations. Phys. Rev. E 55(1), 266 (1997)

    Article  Google Scholar 

  16. Nusse, H.E., Ott, E., Yorke, J.A.: Border-collision bifurcations: an explanation for observed bifurcation phenomena. Phys. Rev. E 49(2), 1073 (1994)

    Article  MathSciNet  Google Scholar 

  17. Piiroinen, P.T., Virgin, L.N., Champneys, A.R.: Chaos and period-adding; experimental and numerical verification of the grazing bifurcation. J. Nonlinear Sci. 14(4), 383–404 (2004)

    Article  MathSciNet  Google Scholar 

  18. Rajasekar, S.: Controlling of chaotic motion by chaos and noise signals in a logistic map and a Bonhoeffer–van der Pol oscillator. Phys. Rev. E 51(1), 775 (1995)

    Article  Google Scholar 

  19. Simpson, D.J., Hogan, S.J., Kuske, R.: Stochastic regular grazing bifurcations. SIAM J. Appl. Dyn. Syst. 12(2), 533–559 (2013)

    Article  MathSciNet  Google Scholar 

  20. Simpson, D.J.W., Kuske, R.: The influence of localized randomness on regular grazing bifurcations with applications to impacting dynamics. J. Vib. Control. 24(2), 407–426

  21. de Souza, S.L., Batista, A.M., Caldas, I.L., Viana, R.L., Kapitaniak, T.: Noise-induced basin hopping in a vibro-impact system. Chaos Solitons Fractals 32(2), 758–767 (2007)

    Article  Google Scholar 

  22. de Souza, S.L., Caldas, I.L., Viana, R.L., Batista, A.M., Kapitaniak, T.: Noise-induced basin hopping in a gearbox model. Chaos Solitons Fractals 26(5), 1523–1531 (2005)

    Article  Google Scholar 

  23. Staunton, E.J., Piiroinen, P.T.: Noise and multistability in the square root map. Physica D: Nonlinear Phenom. 380–381, 31–44 (2018). https://doi.org/10.1016/j.physd.2018.06.002

    Article  MathSciNet  Google Scholar 

  24. Webber, J.B.W.: A bi-symmetric log transformation for wide-range data. Meas. Sci. Technol. 24(2), 027,001 (2012)

    Article  Google Scholar 

  25. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2. Springer, Berlin (2003)

    MATH  Google Scholar 

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Acknowledgements

We are grateful for the hospitality of CRM, Barcelona, where this work was started and to Paul Glendinning for his helpful input in the early stages.

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Authors and Affiliations

  1. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

    Eoghan J. Staunton & Petri T. Piiroinen

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  1. Eoghan J. Staunton
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  2. Petri T. Piiroinen
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Correspondence to Eoghan J. Staunton.

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The authors declare that they have no conflict of interest.

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Eoghan Staunton is supported by an Irish Research Council Postgraduate Scholarship, Award Number GOIPG/2015/3500.

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Staunton, E.J., Piiroinen, P.T. Noise-induced multistability in the square root map. Nonlinear Dyn 95, 769–782 (2019). https://doi.org/10.1007/s11071-018-4595-1

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