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 infinity. This enables the construction of a bifurcation diagram in parameter space showing stable regions where lambda is negative (implying synchronization) and unstable regions where lambda is positive (implying chaotic behavior). The value of lambda depends strongly on the shearing effect of the twist factor of the deterministic Hopf bifurcation. If the twist factor is sufficiently small then lambda is negative regardless of all the other parameters in the system. But when all the parameters except the shear coefficient are fixed then lambda tends to infinity as the shear coefficient tends to infinity.