We investigate the local stability for fixed-depth rotational equatorial flows. We first obtain the precise formula of the second derivative of bifurcation parameters at the bifurcation point. In particular, their signs can be assessed when vorticity is small enough and the mean depth is small or large enough. Furthermore, we study the stability property for periodic traveling waves of the thermocline near the equator with a fixed mean depth. For this aim, we obtain a new stability exchange formula which improves the famous Crandall-Rabinowitz stability exchange theorem.

中文翻译：

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固定深度旋转赤道流非平凡周期性水波的稳定性特性

我们研究了固定深度旋转赤道流的局部稳定性。我们首先得到分岔点处分岔参数二阶导数的精确公式。特别是，当涡度足够小且平均深度足够小或足够大时，可以评估它们的符号。此外，我们研究了赤道​​附近具有固定平均深度的温跃层周期性行波的稳定性特性。为此，我们得到了一个新的稳定性交换公式，该公式改进了著名的 Crandall-Rabinowitz 稳定性交换定理。