Lyapunov exponents and shear-induced chaos for a Hopf bifurcation with additive noise

Abstract This paper considers the effect of additive white noise on the normal form for the supercritical Hopf bifurcation in 2 dimensions. The main results involve the asymptotic behavior of the top Lyapunov exponent λ associated with this random dynamical system as one or more of the parameters in the system tend to 0 or ∞. This enables the construction of a bifurcation diagram in parameter space showing stable regions where λ 0 (implying synchronization) and unstable regions where > 0 λ > 0 (implying chaotic behavior). The value of λ depends strongly on the shearing effect of the twist factor b / a of the deterministic Hopf bifurcation. If b / a is sufficiently small then λ 0 regardless of all the other parameters in the system. But when all the parameters except b are fixed then λ grows like a positive multiple of b^2/3 b 2 / 3 as b b → ∞.