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The Hopf bifurcation with bounded noise

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  • We study Hopf-Andronov bifurcations in a class of random differential equations (RDEs) with bounded noise. We observe that when an ordinary differential equation that undergoes a Hopf bifurcation is subjected to bounded noise then the bifurcation that occurs involves a discontinuous change in the Minimal Forward Invariant set.
    Mathematics Subject Classification: Primary: 37H20; Secondary: 37G10, 34F20.


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  • [1]

    L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.


    L. Arnold, G. Bleckert and K. R. Schenk-Hoppé, The stochastic Brusselator: Parametric noise destroys Hopf bifurcation, in "Stochastic Dynamics" (Bremen, 1997), 71-92, Springer, New York, 1999.


    L. Arnold, N. Sri Namachchivaya and K. R. Schenk-Hoppé, Toward an understanding of stochastic Hopf bifurcation: A case study, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), 1947-1975.doi: 10.1142/S0218127496001272.


    I. Bashkirtseva, L. Ryashko and H. Schurz, Analysis of noise-induced transitions for Hopf system with additive and multiplicative random disturbances, Chaos Solitons Fractals, 39 (2009), 72-82.doi: 10.1016/j.chaos.2007.01.128.


    F. Colonius and W. Kliemann, Topological, smooth, and control techniques for perturbed systems, in "Stochastic Dynamics" (Bremen, 1997) (eds. H. Crauel and M. Gundlach), Springer, New York, (1999), 181-208.


    F. Colonius and W. Kliemann, "The Dynamics of Control," With an appendix by Lars Grüne, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2000.


    J. L. Doob, "Stochastic Processes," John Wiley & Sons, Inc., New York, Chapman & Hall, Limited, London, 1953.


    J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983, revised 1990.


    A. J. Homburg and T. Young, Hard bifurcations in dynamical systems with bounded random perturbations, Regular & Chaotic Dynamics, 11 (2006), 247-258.doi: 10.1070/RD2006v011n02ABEH000348.


    A. J. Homburg and T. Young, Bifurcations for random differential equations with bounded noise on surfaces, Topol. Methods Nonlinear Anal., 35 (2010), 77-97.


    R. A. Johnson, Some questions in random dynamical systems involving real noise processes, in "Stochastic Dynamics" (Bremen, 1997) (eds. H. Crauel and M. Gundlach), Springer, New York, (1999), 147-180.


    Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Applied Mathematical Sciences, 112, Springer Verlag, New York, 1995.


    S. Wieczorek, Stochastic bifurcation in noise-driven lasers and Hopf oscillators, Phys. Rev. E (3), 79 (2009), 036209, 10 pp.


    H. Zmarrou and A. J. Homburg, Bifurcations of stationary measures of random diffeomorphisms, Ergod. Th. Dyn. Systems, 27 (2007), 1651-1692.

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