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).

[2]

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

[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.

[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.

[5]

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

[6]

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

[7]

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

[8]

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

[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.

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

show all references

References:
[1]

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

[2]

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

[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.

[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.

[5]

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

[6]

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

[7]

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

[8]

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

[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.

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

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