We perform a bifurcation analysis of the steady states of Rayleigh--B\'enard convection with no-slip boundary conditions in two dimensions using a numerical method called deflated continuation. By combining this method with an initialisation strategy based on the eigenmodes of the conducting state, we are able to discover multiple solutions to this non-linear problem, including disconnected branches of the bifurcation diagram, without the need for any prior knowledge of the solutions. One of the disconnected branches we find contains an S-shaped curve with hysteresis, which is the origin of a flow pattern that may be related to the dynamics of flow reversals in the turbulent regime. Linear stability analysis is also performed to analyse the steady and unsteady regimes of the solutions in the parameter space and to characterise the type of instabilities.

翻译：我们使用一个称为减缩连续的数值方法,对无滑动边界条件的Rayleigh-B\'enard对齐两个维度的稳态进行双向分析。通过将这种方法与基于进行状态的偏移模式的初始化战略相结合,我们能够找到多种非线性问题的解决方案,包括双向图的断开分支,而无需事先了解解决方案。我们发现的一个断开的分支含有一个带有歇斯底里的S型曲线,这是与动荡状态中流量逆转动态相关的流动模式的起源。还进行了线性稳定性分析,以分析参数空间中解决方案的稳态和不稳性体系,并描述不稳定的类型。