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 \(\lambda \) 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 \(\lambda <0\) (implying synchronization) and unstable regions where \(\lambda > 0\) (implying chaotic behavior). The value of \(\lambda \) depends strongly on the shearing effect of the twist factor b/a of the deterministic Hopf bifurcation. If b/a is sufficiently small then \(\lambda <0\) regardless of all the other parameters in the system. But when all the parameters except b are fixed then \(\lambda \) grows like a positive multiple of \(b^{2/3}\) as \(b \rightarrow \infty \).