Pitchfork and Hopf Bifurcation Under Additive Noise

In this chapter, we study bifurcation of a model of pitchfork bifurcation with additive noise \(\displaystyle \mathrm {d} x= \big (\alpha x-x^3\big )\mathrm {d} t+ \sigma \mathrm {d} W_t\,, \) and a model of Hopf bifurcation with additive noise of the form \(\displaystyle \begin {array}{rl} &\mathrm {d} x = (\alpha x - \beta y - (ax-by)(x^2 + y^2))\,\mathrm {d} t + \sigma \,\mathrm {d} W_t^1\,,\\ &\mathrm {d} y = (\alpha y + \beta x - (bx+ay)(x^2 + y^2))\, \mathrm {d} t + \sigma \,\mathrm {d} W_t^2\,, \end {array} \) near the underlying deterministic bifurcation point \(\alpha =0\) . The bifurcation is characterized by a loss of uniform attractivity of the global random attractor, a change of experimental observability of the Lyapunov exponent, and a change of the dichotomy spectrum of the linearization along the global random attractor. The materials of this part are taken from Callaway et al. (Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 53:1548–1574, 2017), Doan et al. (Nonlinearity 31:4567–4601, 2018).