In the case of dynamo instability, turbulent fluctuations act multiplicatively on the magnetic field, i.e. their effect is proportional to the amplitude of the field. This is referred to as multiplicative noise. Such noise can occur in other contexts. Near the threshold of an instability, fluctuations in the deviation from the threshold will modulate the growth rate of the unstable mode, and this effect appears in the amplitude equation as a multiplicative term. In certain chemical reactions involving two or more species, or more generally when a process requires the presence of two populations, the reaction rate involves the product of the concentrations. If one of the concentrations is kept approximately constant by the operator, fluctuations near the mean value can be described as noise, which will then have a multiplicative effect.
There are numerous studies related to the effect of additive noise on instability. For example, note that Landau's equation for the order parameter of a phase transition at equilibrium can be viewed as the amplitude equation for the unstable mode of a bifurcation. Studies of critical phenomena at equilibrium can thus be transposed to the problem of additive fluctuations near an instability. In general, the effects of additive noise are different from those of multiplicative noise. We focus mostly on the latter case and consider a system that is susceptible to instability in the absence of fluctuations. In the presence of fluctuations, we can ask the following questions:
- How is the instability threshold modified by the fluctuations?
- What happens to the behaviour of the unstable mode above the instability threshold?
- Are any secondary instabilities affected?

Selected publications

   1/f noise and anomalous scaling in Lévy noise-driven on-off intermittency
A. Van Kan, F. Pétrélis,
Journal of Statistical Mechanics, 013204 (2023)

   Growth rate distribution and intermittency in kinematic turbulent dynamos: Which moment predicts the dynamo onset?
K. Seshasayanan and F. Pétrélis
EPL 122 64004 (2018)

   Effect of fluctuations on mean-field dynamos
A. Alexakis, S. Fauve, C. Gissinger, F. Petrelis
J. Plasma Phys, 84, 735840401 (2018)

   Instabilities on a turbulent background
S. Fauve, J. Hérault, G. Michel, F. Pétrélis
Journal of Statistical Mechanics, 064001 (2017)

   Critical Exponents in zero dimensions
A. Alexakis, F. Pétrélis
Journal of Statistical Physics 149, 738-753 (2012)

   Anomalous exponent at the onset of an instability
F. Pétrélis, A. Alexakis
Physical Review Letters (108), 014501, (2012)

   Modification of instability processes by muliplicative noises
Pétrélis F, Aumaitre S,
European Physical Journal B 51 (3), 357-362 (2006)

   Low-frequency noise controls on-off intermittency of bifurcating systems
Aumaître S, Pétrélis F, Mallick K
Physical Review Letters 95 (6), 064101 (2005)

   Effect of phase noise on parametric instabilities
Pétrélis F, Aumaître S, Fauve S
Physical Review Letters 94 (7), 070603 (2005)

   Intermittency at the edge of stochastically inhibited pattern- forming instability
Pétrélis F, Aumaître S
European Physical Journal B, 34 (3), 281-284 (2003)