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 λ 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 λ < 0 (implying synchronization) and unstable regions where λ > 0 λ > 0 (implying chaotic behavior). The value of λ 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 λ < 0 regardless of all the other parameters in the system. But when all the parameters except b are fixed then λ λ grows like a positive multiple of b2/3 b 2 / 3 as b → ∞ b → ∞ .