Finite-time stabilisation by slow-timescale forcing
We make use of a certain one-dimensional stabilisation phenomenon previously described in the physics literature to provide a simple and yet stark illustrative example of how classical mathematical definitions of dynamical stability properties (based on limiting behaviour as time tends to infinity) can be deficient for describing qualitative dynamical stability; and motivated by this example, we develop a new framework for defining and describing stability properties of finite-time systems subject to slowly time-dependent forcing. Namely, our approach is to formulate the dynamics as a slow-fast system in which the slow time is constrained to a compact interval, and define stability properties somewhat analogously to the classical definitions except with the "infinite limit" being in the timescale separation rather than in the infinite progression of time. We obtain rigorous results for the case of one-dimensional systems (which includes our original example). This is joint work with Maxime Lucas and Aneta Stefanovska.
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