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Modeling with Nonsmooth Dynamics

Mike R. Jeffrey
Publication Date: 
Number of Pages: 
Frontiers in Applied Dynamical Systems: Reviews and Tutorials
[Reviewed by
Bill Satzer
, on
Nonsmooth dynamics” is a term used to describe a system where the equations describing its behavior have abrupt changes at certain thresholds. The abruptness of the change occurs on such a short time scale that it is treated as a discontinuity. (For example, think of the change in velocity during the collision of a fast moving ball hitting a wall and its rebound.) Discontinuities like this can arise whenever a system incorporates physical switches, changes in modes of contact between components, encounters with boundaries of solid objects, or even decisions by a human or an electronic controller.
One reason why handling discontinuities in dynamical systems is difficult is that they arise because of incomplete knowledge of the underlying processes that occur when an abrupt transition occurs. In the last few years, interest in nonsmooth systems has grown substantially. Engineers, control theorists and others have dealt with these questions for some time – with differential equations that have discontinuous right-hand sides, for example. Yet the first significant work by mathematicians began around 1960 with work by Russian mathematicians. Filippov’s Differential Equations with Discontinuous Righthand Sides first represented and formalized some of this earlier work by the Russians. 
By and large the powerful results of differentiable dynamical systems theory - stability, bifurcation, the structure of attractors – do not apply to nonsmooth systems. New formulations are needed for systems that are only piecewise smooth. But it isn’t just the loss of smoothness that causes discomfort. What’s worse is that when discontinuous quantities appear in differential equations, non-uniqueness also pops up.
The approach that Filippov described shows that solutions to discontinuous problems exist and are as amenable to analysis as smooth systems. His work formalized the idea of a “sliding surface”, for sliding along a discontinuity. For a very simple example, consider a dynamical system in the plane with a line of discontinuities that separates two regions with smooth vector fields. A solution can start in one region, reach the sliding surface and move along it for a time, and then cross into the other region. This is one place where non-uniqueness can arise.
This short monograph does not attempt to present an all-encompassing theory. Instead, its goal is to introduce a framework somewhat broader than Filippov’s that is capable of addressing dynamical systems with discontinuities and can provide a path for a future consistent theory of piecewise smooth dynamics. The author uses examples that illustrate the current state of understanding the seemingly ambiguous behavior that can occur in nonsmooth systems. These include a two-gene regulatory system with apparently ambiguous activation of genes and an investment game with two players where their seemingly steady behavior destabilizes the trading of a company’s stock.
The author argues that a few modifications to Filippov’s original formulation can move forward both theory and applications of nonsmooth systems. He suggests some appropriate generalizations that introduce nonlinearity into Filippov’s idealized conception. More specifically this means turning the sliding surface into a sliding layer, introducing sliding attractors to replace sliding modes, and using nonlinear switching as a general model of dynamics at a discontinuity.
This book is best matched to graduate students and researchers with a background in basic dynamical systems. It has an extensive bibliography and would provide a great source for new research problems. 
Bill Satzer (, now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films and ceramic fiber-reinforced composites. Along the way he learned more about ceramics and alloys of aluminum than he could have imagined in graduate school. He did his PhD work in dynamical systems.