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