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We prove the positivity of Lyapunov exponents for the normal form of a Hopf bifurcation, perturbed by additive white noise, under sufficiently strong shear strength. This completes a series of related results for simplified situations which we can exploit by studying suitable limits of the shear and noise parameters. The crucial technical ingredient for making this approach rigorous is a result on the continuity of Lyapunov exponents via Furstenberg–Khasminskii formulas. [ABSTRACT FROM AUTHOR]

We introduce the notion of Lyapunov exponents for random dynamical systems, conditioned to trajectories that stay within a bounded domain for asymptotically long times. This is motivated by the desire to characterize local dynamical properties in the presence of unbounded noise (when almost all trajectories are unbounded). We illustrate its use in the analysis of local bifurcations in this context. The theory of conditioned Lyapunov exponents of stochastic differential equations builds on the stochastic analysis of quasi-stationary distributions for killed processes and associated quasi-ergodic distributions. We show that conditioned Lyapunov exponents describe the asymptotic stability behaviour of trajectories that remain within a bounded domain and, in particular, that negative conditioned Lyapunov exponents imply local synchronisation. Furthermore, a conditioned dichotomy spectrum is introduced, and its main characteristics are established. [ABSTRACT FROM AUTHOR]

We establish the existence of a bifurcation from an attractive random equilibrium to shear-induced chaos for a stochastically driven limit cycle, indicated by a change of sign of the first Lyapunov exponent. This relates to an open problem posed by Lin and Young (Nonlinearity 21:899-922, 2008) and Young (Nonlinearity 21:245-252, 2008), extending results by Wang and Young (Commun Math Phys 240(3):509-529, 2003) on periodically kicked limit cycles to the stochastic context. [ABSTRACT FROM AUTHOR]