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The Stochastic Brusselator: Parametric Noise Destroys Hoft Bifurcation

In: Stochastic Dynamics

Author

Listed:
  • Ludwig Arnold
  • Gabriele Bleckert
  • Klaus Reiner Schenk-Hoppé

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.

Suggested Citation

  • Ludwig Arnold & Gabriele Bleckert & Klaus Reiner Schenk-Hoppé, 1999. "The Stochastic Brusselator: Parametric Noise Destroys Hoft Bifurcation," Springer Books, in: Stochastic Dynamics, chapter 4, pages 71-92, Springer.
    • Handle: RePEc:spr:sprchp:978-0-387-22655-2_4
    DOI: 10.1007/0-387-22655-9_4
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