  Magnetorotational dynamo action in Keplerian shear flow is a
three-dimensional, nonlinear magnetohydrodynamic process whose study is
relevant to the understanding of accretion and magnetic field generation in
astrophysics. Transition to this form of dynamo is subcritical and shares many
characteristics of transition to turbulence in non-rotating hydrodynamic shear
flows. This suggests that these different fluid systems become active through
similar generic bifurcation mechanisms, which in both cases have eluded
detailed understanding so far. In this paper, we investigate numerically the
bifurcation mechanisms at work in the incompressible Keplerian
magnetorotational dynamo problem in the shearing box framework. Using numerical
techniques imported from dynamical systems research, we show that the onset of
chaotic dynamo action at magnetic Prandtl numbers larger than unity is
primarily associated with global homoclinic and heteroclinic bifurcations of
nonlinear magnetorotational dynamo cycles. These global bifurcations are
supplemented by local bifurcations of cycles marking the beginning of
period-doubling cascades. This suggests that nonlinear magnetorotational dynamo
cycles provide the pathway to turbulent injection of both kinetic and magnetic
energy in incompressible magnetohydrodynamic Keplerian shear flow in the
absence of an externally imposed magnetic field. Studying the nonlinear physics
and bifurcations of these cycles in different regimes and configurations may
subsequently help to better understand the conditions of excitation of
magnetohydrodynamic turbulence and instability-driven dynamos in various
astrophysical systems and laboratory experiments. The detailed characterization
of global bifurcations provided for this three-dimensional subcritical fluid
dynamics problem may also prove useful for the problem of transition to
turbulence in hydrodynamic shear flows.
