For general nonlinear systems the the dynamics is differentiable but for fast switching system this may not hold.
InsyAllah I will explain in detail soon.
You can have a look on the work of Mario di Bernardo. If you find different approaches, please lend me know, I am working also on contraction theory.
https://www.researchgate.net/publication/280538408_Contraction_analysis_of_switched_Filippov_systems_via_regularization
Article Contraction analysis of switched Filippov systems via regularization
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