Abstract
We perform mainly a numerical study of the bifurcation behavior of the Brusselator under parametric white noise. It was shown before that parametric noise turns the deterministic Hopf bifurcation into a scenario in which the stationary density (unique solution of the Fokker- Planck equation) undergoes a delayed transition from a single-peaked, bellshaped to a crater-type form. We will make this more precise by showing that the stationary density gets a “dent” at the deterministic bifurcation point and develops a local minimum at a later parameter value. In contrast (but not in contradiction) to these findings we will show that, from the view point of random dynamical systems, the deterministic Hopf bifurcation is being “destroyed” by parametric noise in the following sense: For all values of the bifurcation parameter, the system has a unique invariant measure which is, moreover, exponentially stable in the sense that its top Lyapunov exponent is negative. The invariant measure is a random Dirac measure, and its support is the global random attractor of the system.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
5 References
V. Altares and G. Nicolis. Stochastically forced Hopf bifurcation: approximate Fokker Planck equation in the limit of short correlation times. Physical Review A, 37:3630–3633, 1988.
L. Arnold. On the consistency of the mathematical models of chemical reactions. In H. Haken, editor, Dynamics of Synergetic Systems, volume 6 of Series in Synergetics, pages 107–118. Springer-Verlag, Berlin, 1980.
L. Arnold. Six lectures on random dynamical systems. In R. Johnson, editor, Dynamical Systems (CIME Summer School 1994), volume 1609 of Lecture Notes in Mathematics, pages 1–43. Springer-Verlag, Berlin, 1995.
L. Arnold. Random dynamical systems. Springer-Verlag, Berlin, 1998.
L. Arnold and P. Imkeller. Rotation numbers for linear stochastic differential equations. Report 415, Institut für Dynamische Systeme, Universität Bremen, 1997.
F. Baras. Stochastic analysis of limit cycle behavior. In L. Schimansky-Geier and T. Pöschel, editors, Stochastic Dynamics, volume 484 of Lecture Notes in Physics, pages 167–178. Springer-Verlag, Berlin, 1997.
G. Bleckert and K. R. Schenk-Hoppé. Software for a numerical study of the stochastic Brusselator, 1998. Can be obtained at the URL http://www.wiwi.uni-bielefeld.de/~boehm/members/klaus/brusselator/.
F. Colonius and W. Kliemann. Random perturbations of bifurcation diagrams. Nonlinear Dynamics, 5:353–373, 1994.
H. Crauel and F. Flandoli. Additive noise destroys a pitchfork bifurcation. Journal of Dynamics and Differential Equations, 10:259–274, 1998.
M. Ehrhardt. Invariant probabilities for systems in a random environment — with applications to the Brusselator. Bulletin of Mathematical Biology, 45:579–590, 1983.
L. Fronzoni, R. Mannella, P. V. E. McClintock, and F. Moss. Postponement of Hopf bifurcations by multiplicative colored noise. Physical Review A, 36:834–841, 1987.
W. Horsthemke and R. Lefever. Noise-induced transitions. Springer-Verlag, Berlin, 1984.
W. Kliemann. Recurrence and invariant measures for degenerate diffusions. The Annals of Probability, 15:690–707, 1987.
M. Krebs. Bifurkationsverhalten des stochastischen Brusselators. Diplomarbeit, Universität Bremen, 1995.
R. Lefever and G. Nicolis. Chemical instabilities and sustained oscillations. Journal of Theoretical Biology, 30:267–284, 1971.
R. Lefever and J. Turner. Sensitivity of a Hopf bifurcation to external multiplicative noise. In W. Horsthemke and D. K. Kondepudi, editors, Fluctuations and sensitivity in nonequilibrium systems, pages 143–149. Springer-Verlag, Berlin, 1984.
R. Lefever and J. Turner. Sensitivity of a Hopf bifurcation to multiplicative colored noise. Physical Review Letters, 56:1631–1634, 1986.
G. Leng, N. Sri Namachchivaya, and S. Talwar. Robustness of nonlinear systems perturbed by external random excitation. ASME Journal of Applied Mechanics, 59:1–8, 1992.
M. Malek Mansour, C. van den Broeck, G. Nicolis, and J. W. Turner. Asymptotic properties of Markovian Master equations. Annals of Physics, 131:283–313, 1981.
F. Moss and P. V. E. McClintock. Noise in nonlinear dynamical systems, Volumes 1—3. Cambridge University Press, 1989.
G. Nicolis and I. Prigogine. Self-organization in non-equilibrium systems. Wiley, New York, 1977.
P. J. Ponzo and N. Wax. Note on a model of a biochemical reaction. Journal of Mathematical Analysis and Applications, 66:354–357, 1978.
K. R. Schenk-Hoppé. Deterministic and stochastic Duffing-van der Pol oscillators are non-explosive. ZAMP-Journal of Applied Mathematics and Physics, 47:740–759, 1996.
H. Sussmann. On the gap between deterministic and stochastic ordinary differential equations. The Annals of Probability, 6:19–41, 1978.
J. Tyson. Some further studies of nonlinear oscillations in chemical systems. Journal of Chemical Physics, 58:3919–3930, 1973.
Ye Yan-Qian. Theory of limit cycles. American Mathematical Society, Providence, Rhode Island, 1986.
Rights and permissions
Copyright information
© 1999 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Arnold, L., Bleckert, G., Schenk-Hoppé, K.R. (1999). The Stochastic Brusselator: Parametric Noise Destroys Hoft Bifurcation. In: Stochastic Dynamics. Springer, New York, NY. https://doi.org/10.1007/0-387-22655-9_4
Download citation
DOI: https://doi.org/10.1007/0-387-22655-9_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-98512-1
Online ISBN: 978-0-387-22655-2
eBook Packages: Springer Book Archive