Dynamics and bifurcations of nonsmooth systems: A survey

By Oleg Makarenkov, Jeroen S. W. Lamb. 2012

321 citations recorded by the source corpus.

Abstract In this survey we discuss current directions of research in the dynamics of nonsmooth systems, with emphasis on bifurcation theory. An introduction to the state-of-the-art (also for non-specialists) is complemented by a presentation of the main open problems. We illustrate the theory by means of elementary examples. The main focus is on piecewise smooth systems, which have recently attracted a lot of attention, but we also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities. This extended framework allows us to put the diverse range of papers and surveys in this special issue in a common context. A dedicated section is devoted to concrete applications that stimulate the development of the field. This survey is concluded by an extensive bibliography.

Subfields: Control and Systems Engineering, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/16ab3fdc87ca60cedb2ef48fc771cb07c3e55cee

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In Control and Systems Engineering

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Citations in the corpus

On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line (2014, background): “, see for instance the books of di Bernardo, Budd, Champneys and Kowalczyk [3], and Simpson [25], and the survey of Makarenkov and Lamb [23], and the hundreds of references quoted in these last three works.”
Limit cycles for discontinuous planar piecewise linear differential systems separated by one straight line and having a center (2018, background): “Many phenomena of the real world are related with the existence of limit cycles, some examples are the van der Pol oscillator [28, 29], or the Belousov–Zhabotinskii reaction [3, 31], and many other examples can be found in the book [4], or in the survey [22].”
Piecewise linear differential systems with only centers can create limit cycles? (2017, background): “[1] and until nowadays they had special attention from themathematicians, mainly because these systems appear in mechanics, electrical circuits, economy, etc, see for instance the books [6,27] and the surveys [23,28].”