Multiplicative noise creates new phases absent in the deterministic system
Mode: unimodal
Horsthemke-Lefever noise-induced transition: dX = (aX − X³)dt + D·X·dW.
The multiplicative noise term D·X·dW modifies the stationary distribution P(x) ∝ |x|^(2a/D²−1) · exp(−x²/D²).
At the critical noise D_c = √(2a), P(x) transitions from unimodal (single peak at x=0 or x≠0)
to bimodal — a new phase emerges purely from noise, with no deterministic counterpart.
Histogram shows 500 Euler-Maruyama trajectories; colored line shows analytic P(x).