This manuscript aims to analyze the chaotic fluid layer heating from below, using the Atangana–Baleanu derivative to characterize the existence and uniqueness of this chaotic model based on the provided fixed point data. First, with suitable examples, we demonstrate the existence and uniqueness of fixed point solutions within the framework of controlled metric spaces. Secondly, we study the chaotic behavior of water with various Rayleigh numbers by experimenting with it while keeping a constant Prandtl number. Furthermore, the role of the Rayleigh number in magnifying chaotic behaviors is examined, with graphical simulations demonstrating the effect of different fractional orders of the Atangana–Baleanu derivative. This research provides insights into the dynamics of chaotic systems and their practical applications in controlled situations.