Dynamical bifurcation with noise
- 59 Downloads
It was shown by A. Neishtadt that dynamical bifurcation, in which the control parameter is varied with a small but finite speed ∈, is characterized by adelay in bifurcation, here denoted λj and depending on ∈. Here we study dynamical bifurcation, in the framework and with the language of Landau theory of phase transitions, in the presence of a Gaussian noise of strength σ. By numerical experiments at fixed ∈ = ∈0, we study the dependence of λj on a for order parameters of dimension ≤3; an exact scaling relation satisfied by the equations permits us to obtain for this the behavior for general ∈. We find that in the smallnoise regime λj(σ) ≃aσ(−b), while in the strong-noise regime λj(σ) − ce(−d); we also measure the parameters in these formulas.
KeywordsPhase Transition Field Theory Elementary Particle Quantum Field Theory Numerical Experiment
Unable to display preview. Download preview PDF.
- Aframovich, V. S., Arnold, V. I., Ilyashenko, Yu. S., and Shilnikov, L. P. (1988). Theory of bifurcations, inDynamical Systems V, Encyclopedia of Mathematical Science, Springer, Berlin.Google Scholar
- Benoit, E., ed. (1991).Dynamic Bifurcations, Springer, Berlin.Google Scholar
- Fronzoni, L., Moss, F., and McClintock, P. V. E. (1987).Physical Review A,36, 1492.Google Scholar
- Gaeta, G. (1993). Dynamical bifurcations and competing instabilities in Landau and Landau-Ginzburg theory, preprint IFU-Roma.Google Scholar
- Landau, L. D., and Lifshitz, E. M. (1958).Statistical Mechanics, Pergamon, London.Google Scholar
- Lobry, C. (1991). InDynamic Bifurcations, E. Benoit, ed., Springer, Berlin.Google Scholar
- Lythe, G. D., and Proctor, M. R. E. (1993).Physical Review E,47, 3122.Google Scholar
- Neishtadt, A. (1988a).Differential Equations,23, 1385.Google Scholar
- Neishtadt, A. (1988b).Differential Equations,24, 171.Google Scholar
- Torrent, M. C., and San Miguel, M. (1988).Physical Review A,38, 245.Google Scholar
- Van der Broek, C., and Mandel, P. (1987).Physics Letters A,122, 36.Google Scholar