The phenomenon produced by the Hopf bifurcations is of notable importance. In fact, a Hopf bifurcation—guaranteeing the existence of an unsteady periodic solution of the linearized problem at stake—is also an optimum limit cycle candidate of the nonlinear associated problem and, if non linearly globally attractive, is an absorbing set and an effective limit cycle. The present paper deals with the onset of Hopf bifurcations in thermal magnetohydrodynamics (MHD). Precisely, it is devoted to characterization—via a simple formula—of the Hopf bifurcations threshold in horizontal plasma layers between rigid planes, heated from below and embedded in a constant transverse magnetic field. This problem, remarked clearly and notably by the Nobel Laureate Chandrasekhar (Nature 175:417–419, 1955), constitutes a difficulty met by him and—for plasma layers between rigid planes electricity perfectly conducting—is, as far as we know, still not removed. Let \(m_0\) be the thermal conduction rest state and let \(P_r, P_m, R, Q\), be the Prandtl, the Prandtl magnetic, the Rayleigh and the Chandrasekhar number, respectively. Recognized (according to Chandrasekhar) that the instability of \(m_0\) via Hopf bifurcation can occur only in a plasma with \(P_m>P_r\), in this paper it is shown that the Hopf bifurcations occur if and only if

with \( \mu =7.8532\). Moreover, the critical value of R at which the Hopf bifurcation occurs is characterized via the smallest zero of the second invariant of the spectrum equation governing the most destabilizing perturbation. The critical value of Q, in the free-rigid and rigid-free cases is shown to be \(\displaystyle \frac{1}{4}\) of the previous value.

Hopf分叉产生的现象非常重要。实际上，Hopf分叉-保证线性化问题的非定常周期解的存在-也是非线性相关问题的最佳极限环候选者，并且如果不是非线性全局吸引子，则是一个吸收集和有效极限周期。本文讨论了热磁流体动力学（MHD）中霍普夫分叉的发生。精确地，它致力于通过一个简单的公式表征刚性平面之间的水平等离子层中的Hopf分叉阈值，这些阈值从下方加热并嵌入恒定的横向磁场中。诺贝尔奖获得者钱德拉塞卡（Chandrasekhar）（自然，175：417–419，1955年）清楚地指出了这一问题，这就构成了他遇到的困难，并且-就刚性平面之间的等离子体层而言，电的完美传导-据我们所知仍未被消除。让\（m_0 \）是热传导静止状态，而\（P_r，P_m，R，Q \）分别是普朗特，普朗特磁，瑞利和钱德拉塞卡数。根据钱德拉塞卡（Chandrasekhar）认识到，通过Hopf分叉产生的\（m_0 \）的不稳定性只能在具有\（P_m> P_r \）的等离子体中发生，本文证明了Hopf分叉发生于且仅当

与\（\ mu = 7.8532 \）。此外，发生霍普夫分叉的R的临界值通过支配最不稳定的扰动的光谱方程第二不变量的最小零来表征。在自由刚性和自由自由的情况下，Q的临界值显示为先前值的\（\ displaystyle \ frac {1} {4} \）。