August  2012, 32(8): 2997-3007. doi: 10.3934/dcds.2012.32.2997

The Hopf bifurcation with bounded noise

1. 

Department of Mathematical, Information & Computer Sciences, Point Loma Nazarene University, 3900 Lomaland Drive, San Diego, CA 92106, United States

2. 

KdV Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH, Amsterdam, Netherlands

3. 

Department of Mathematics, Ohio University, 321 Morton Hall, OH 45701 Athens, United States

Received  May 2011 Revised  July 2011 Published  March 2012

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.
Citation: Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997
References:
[1]

L. Arnold, "Random Dynamical Systems,", Springer Monographs in Mathematics, (1998).   Google Scholar

[2]

L. Arnold, G. Bleckert and K. R. Schenk-Hoppé, The stochastic Brusselator: Parametric noise destroys Hopf bifurcation,, in, (1997), 71.   Google Scholar

[3]

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.  doi: 10.1142/S0218127496001272.  Google Scholar

[4]

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.  doi: 10.1016/j.chaos.2007.01.128.  Google Scholar

[5]

F. Colonius and W. Kliemann, Topological, smooth, and control techniques for perturbed systems,, in, (1999), 181.   Google Scholar

[6]

F. Colonius and W. Kliemann, "The Dynamics of Control," With an appendix by Lars Grüne,, Systems & Control: Foundations & Applications, (2000).   Google Scholar

[7]

J. L. Doob, "Stochastic Processes,", John Wiley & Sons, (1953).   Google Scholar

[8]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Applied Mathematical Sciences, 42 (1983).   Google Scholar

[9]

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

[10]

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

[11]

R. A. Johnson, Some questions in random dynamical systems involving real noise processes,, in, (1999), 147.   Google Scholar

[12]

Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", Applied Mathematical Sciences, 112 (1995).   Google Scholar

[13]

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

[14]

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

show all references

References:
[1]

L. Arnold, "Random Dynamical Systems,", Springer Monographs in Mathematics, (1998).   Google Scholar

[2]

L. Arnold, G. Bleckert and K. R. Schenk-Hoppé, The stochastic Brusselator: Parametric noise destroys Hopf bifurcation,, in, (1997), 71.   Google Scholar

[3]

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.  doi: 10.1142/S0218127496001272.  Google Scholar

[4]

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.  doi: 10.1016/j.chaos.2007.01.128.  Google Scholar

[5]

F. Colonius and W. Kliemann, Topological, smooth, and control techniques for perturbed systems,, in, (1999), 181.   Google Scholar

[6]

F. Colonius and W. Kliemann, "The Dynamics of Control," With an appendix by Lars Grüne,, Systems & Control: Foundations & Applications, (2000).   Google Scholar

[7]

J. L. Doob, "Stochastic Processes,", John Wiley & Sons, (1953).   Google Scholar

[8]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Applied Mathematical Sciences, 42 (1983).   Google Scholar

[9]

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

[10]

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

[11]

R. A. Johnson, Some questions in random dynamical systems involving real noise processes,, in, (1999), 147.   Google Scholar

[12]

Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", Applied Mathematical Sciences, 112 (1995).   Google Scholar

[13]

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

[14]

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

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