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

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 $$\lambda $$ λ associated with this random dynamical system as one or more of the parameters in the system tend to 0 or $$\infty $$ ∞ . This enables the construction of a bifurcation diagram in parameter space showing stable regions where $$\lambda <0$$ λ < 0 (implying synchronization) and unstable regions where $$\lambda > 0$$ λ > 0 (implying chaotic behavior). The value of $$\lambda $$ 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 $$\lambda <0$$ λ < 0 regardless of all the other parameters in the system. But when all the parameters except b are fixed then $$\lambda $$ λ grows like a positive multiple of $$b^{2/3}$$ b 2 / 3 as $$b \rightarrow \infty $$ b → ∞ .