  We study a class of bifurcations generically occurring in dynamical systems
with non-mutual couplings ranging from models of coupled neurons to
predator-prey systems and non-linear oscillators. In these bifurcations,
extended attractors such as limit cycles, limit tori, and strange attractors
merge and split in a similar way as fixed points in a pitchfork bifurcation. We
show that this merging and splitting coincides with the coalescence of
covariant Lyapunov vectors with vanishing Lyapunov exponents, generalizing the
notion of exceptional points to non-linear dynamical systems. We distinguish
two classes of bifurcations, corresponding respectively to continuous and
discontinuous behaviors of the covariant Lyapunov vectors at the transition. We
outline some physical consequences of generalized exceptional points on the
dynamics of the system, including non-reciprocal responses, the destruction of
isochrons, and enhanced sensitivity to noise. We illustrate our results with
concrete examples from neuroscience, ecology, and physics. When applied to
interpret existing experimental observations, our analysis suggests a simple
explanation for the non-trivial phase delays observed in the population
dynamics of plankton communities and the recently measured statistics of
rotation reversals for a solid body immersed in a Rayleigh-B\'enard convection
cell.
