Beyond the Bristol book: advances and perspectives in non-smooth dynamics and applications. (English) Zbl 1541.34031
Summary: Non-smooth dynamics induced by switches, impacts, sliding, and other abrupt changes are pervasive in physics, biology, and engineering. Yet, systems with non-smooth dynamics have historically received far less attention compared to their smooth counterparts. The classic “Bristol book” [M. di Bernardo et al., Piecewise-smooth dynamical systems. Theory and applications. New York, NY: Springer (2008; Zbl 1146.37003)] contains a 2008 state-of-the-art review of major results and challenges in the study of non-smooth dynamical systems. In this paper, we provide a detailed review of progress made since 2008. We cover hidden dynamics, generalizations of sliding motion, the effects of noise and randomness, multi-scale approaches, systems with time-dependent switching, and a variety of local and global bifurcations. Also, we survey new areas of application, including neuroscience, biology, ecology, climate sciences, and engineering, to which the theory has been applied.
MSC:
| 34A36 | Discontinuous ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 35B32 | Bifurcations in context of PDEs |
| 34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |
| 37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |
Citations:
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