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This work studies two-dimensional fixed-flux Rayleigh–Bénard convection with periodic boundary conditions in both horizontal and vertical directions and analyses 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 behaviour between this state and a time-dependent direction-reversing state, followed by a global bifurcation leading to modulated travelling waves without flow reversal. Single-mode equations reproduce this moderate Rayleigh number behaviour well. At high Rayleigh numbers, chaotic behaviour dominated by modulated travelling 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 quasi-periodic and chaotic behaviour as the Rayleigh number increases. In the high$$Pr$$regime, the large-scale shear weakens, and the flow shows bursting behaviour that can lead to significantly increased heat transport or even intermittent stable stratification.