Critical exponents are calculated exactly at the onset of an instability, by using asymptotic expansion techniques. When the unstable mode is subject to multiplicative noise whose spectrum at zero frequency vanishes, we show that the critical behavior can be anomalous; i.e., the mode amplitude $X$ scales with departure from onset $\mu$ as $⟨X⟩\propto {\mu }^{\beta }$ with an exponent $\beta$ different from its deterministic value. This behavior is observed in a direct numerical simulation of the dynamo instability, and our results provide a possible explanation for recent experimental observations.

• Received 12 October 2011

DOI:https://doi.org/10.1103/PhysRevLett.108.014501