We perform a bifurcation analysis of a two-dimensional magnetic Rayleigh--B\'enard problem using a numerical technique called deflated continuation. Our aim is to study the influence of the magnetic field on the bifurcation diagram as the Chandrasekhar number $Q$ increases and compare it to the standard (non-magnetic) Rayleigh--B\'enard problem. We compute steady states at a high Chandrasekhar number of $Q=10^3$ over a range of the Rayleigh number $0\leq \Ra\leq 10^5$. These solutions are obtained by combining deflation with a continuation of steady states at low Chandrasekhar number, which allows us to explore the influence of the strength of the magnetic field as $Q$ increases from low coupling, where the magnetic effect is almost negligible, to strong coupling at $Q=10^3$. We discover a large profusion of states with rich dynamics and observe a complex bifurcation structure with several pitchfork, Hopf, and saddle-node bifurcations. Our numerical simulations show that the onset of bifurcations in the problem is delayed when $Q$ increases, while solutions with fluid velocity patterns aligning with the background vertical magnetic field are privileged. Additionally, we report a branch of states that stabilizes at high magnetic coupling, suggesting that one may take advantage of the magnetic field to discriminate solutions.

翻译：我们采用一种称为"收缩延拓"的数值技术，对二维磁瑞利-贝纳尔问题进行了分岔分析。本研究旨在探讨随着钱德拉塞卡数$Q$增大，磁场对分岔图的影响，并将其与标准（非磁性）瑞利-贝纳尔问题进行对比。我们在钱德拉塞卡数$Q=10^3$的高磁耦合条件下，计算了瑞利数$0\leq \Ra\leq 10^5$范围内的稳态解。这些解是通过将收缩算法与低钱德拉塞卡数下的稳态延拓相结合而获得的，该方法使我们能够系统研究磁场强度的影响——从磁效应几乎可忽略的弱耦合状态，到$Q=10^3$的强耦合状态。我们发现了大量具有丰富动力学特性的状态，并观测到包含多个叉形分岔、霍普夫分岔和鞍结分岔的复杂分岔结构。数值模拟表明：随着$Q$增大，该问题的分岔起始点会出现延迟，同时流体速度模式与背景垂直磁场方向对齐的解会占据主导地位。此外，我们报告了一个在高磁耦合条件下趋于稳定的解分支，这表明可以利用磁场对解进行筛选。