  We explore the instabilities developed in a fluid in which viscosity depends
on temperature. In particular, we consider a dependency that models a very
viscous (and thus rather rigid) lithosphere over a convecting mantle. To this
end, we study a 2D convection problem in which viscosity depends on temperature
by abruptly changing its value by a factor of 400 within a narrow temperature
gap. We conduct a study which combines bifurcation analysis and time-dependent
simulations. Solutions such as limit cycles are found that are fundamentally
related to the presence of symmetry. Spontaneous plate-like behaviors that
rapidly evolve towards a stagnant lid regime emerge sporadically through abrupt
bursts during these cycles. The plate-like evolution alternates motions towards
either the right or the left, thereby introducing temporary asymmetries on the
convecting styles. Further time-dependent regimes with stagnant and plate-like
lids are found and described.
