Finite-time coherent sets represent minimally mixing objects in general nonlinear dynamics, and are spatially mobile features that are the most predictable in the medium term. When the dynamical system is subjected to small parameter change, one can ask about the rate of change of (i) the location and shape of the coherent sets, and (ii) the mixing properties (how much more or less mixing), with respect to the parameter. We answer these questions by developing linear response theory for the eigenfunctions of the dynamic Laplace operator, from which one readily obtains the linear response of the corresponding coherent sets. We construct efficient numerical methods based on a recent finite-element approach and provide numerical examples.

有限时间相干集代表一般非线性动力学中最小混合的对象，并且是在中期最可预测的空间移动特征。当动力系统受到很小的参数变化时，人们可以询问（i）相干集合的位置和形状，以及（ii）混合特性（混合多少或多少）的变化率，关于到参数。我们通过发展动态拉普拉斯算子的本征函数的线性响应理论来回答这些问题，从中可以很容易地获得相应相干集的线性响应。我们基于最近的有限元方法构建了有效的数值方法并提供了数值示例。