About a fixed point the action of a continuous map can be decomposed into a rotation and an expansion. Here it is shown how the stability of a fixed point can be determined by averaging the expansion over invariant measures of the rotation. This is useful for fixed points on switching manifolds of piecewise-smooth maps where stability cannot be determined in the usual way from the eigenvalues of a Jacobian matrix. The results are applied to a model of a forced oscillator subject to stick–slip friction and provide a simple and novel approximation to the parameter regime within which a grazing-sliding bifurcation creates an attractor. The results are further explored with a two-dimensional, non-invertible, piecewise-linear map for which the fixed point is stable in an extremely complicated subset of parameter space. This is explained by the fact that the rotation is chaotic and has an infinity of invariant measures. A measure-theoretic notion of stability is introduced and used in addition to standard notions of stability because the fixed point can be both a Milnor attractor and unstable.

中文翻译：

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分段光滑连续映射的切换流形上的不动点的稳定性

围绕固定点，连续贴图的作用可以分解为旋转和扩展。此处显示了如何通过平均旋转不变量的扩展来确定固定点的稳定性。这对于分段平滑映射的切换歧管上的固定点很有用，在这些固定点上，无法以通常的方式根据雅可比矩阵的特征值确定稳定性。将结果应用于受粘滑摩擦的强迫振荡器模型，并为参数制度提供了一种简单新颖的近似方法，在该参数制度中，掠过滑动的分叉产生了一个吸引子。通过二维不可逆分段线性映射进一步探索结果，对于该映射，固定点在极其复杂的参数空间子集中是稳定的。这是由于旋转是混沌的并且具有无穷大的不变量这一事实来解释的。由于固定点既可以是Milnor吸引子又可以是不稳定的，因此除了标准的稳定性概念之外，还引入并使用了量度理论的稳定性概念。