Linear response for the dynamic Laplacian and finite-time coherent sets

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.

Automated summary

Finite-time coherent sets are minimally mixing objects in general nonlinear dynamics, with predictable spatial mobility in the medium term. This study provides answers regarding the rate of change and mixing properties of coherent sets under small parameter changes, through the development of linear response theory for the eigenfunctions of the dynamic Laplace operator. Efficient numerical methods based on a recent finite-element approach are constructed and numerical examples are provided.