  This work studies two-dimensional fixed-flux Rayleigh-B\'enard convection
with periodic boundary conditions in both horizontal and vertical directions
and analyzes its dynamics using numerical continuation, secondary instability
analysis and direct numerical simulation. The fixed-flux constraint leads to
time-independent elevator modes with a well-defined amplitude. Secondary
instability of these modes leads to tilted elevator modes accompanied by
horizontal shear flow. For $Pr$=1, where $Pr$ is the Prandtl number, a
subsequent subcritical Hopf bifurcation leads to hysteresis behavior between
this state and a time-dependent direction-reversing state, followed by a global
bifurcation leading to modulated traveling waves without flow reversal.
Single-mode equations reproduce this moderate Rayleigh number behavior well. At
high Rayleigh numbers, chaotic behavior dominated by modulated traveling waves
appears. These transitions are characteristic of high wavenumber elevator modes
since the vertical wavenumber of the secondary instability is linearly
proportional to the horizontal wavenumber of the elevator mode. At a low $Pr$,
relaxation oscillations between the conduction state and the elevator mode
appear, followed by quasiperiodic and chaotic behavior as the Rayleigh number
increases. In the high $Pr$ regime, the large-scale shear weakens, and the flow
shows bursting behavior that can lead to significantly increased heat transport
or even intermittent stable stratification.
