A mathematical framework for critical transitions: Bifurcations, fast-slow systems and stochastic dynamics

► We give an overview of dynamical systems theory applicable to critical transitions. ► A rigorous definition and many results follow from this framework. ► We investigate the variance as an early-warning sign using normal forms. ► Numerical, geometric and analytical results for the variance are obtained. ► Stochastic P-bifurcations are suggested as a new early-warning sign.