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Bifurcations in nonsmooth dynamical systems. (English) Zbl 1168.34006

The authors consider dynamical systems that are piecewise smooth. For this type of dynamical systems, the phase space is partitioned into regions by smooth hypersurfaces. In each region the system is governed by a smooth ordinary differential equation but at the boundaries (that is, between regions) discontinuities may occur. The paper distinguishes three classes of systems, depending on the type of discontinuity that occurs on the boundaries: a jump of the trajectory in the phase space (this happens, for example, at an idealized mechanical impact), a jump of the right-hand-side (for example, due to an electrical switch, or dry friction), or a discontinuity in a (higher) derivative of the right-hand-side (caused, for example, by a spring with backlash).
The authors then study discontinuity induced bifurcations (DIBs) of codimension one for each of these classes, defining a DIB as a nontrivial interaction of a limit set with a boundary. Limit sets considered in this review are equilibria and limit cycles, and the interaction has to be local (that is, at a single point of the boundary): for example, an equilibrium collides with a boundary, or a limit cycle grazes a boundary tangentially (or hits the intersection of two boundaries) under variation of a system parameter. Depending on the class various phenomena may occur, for example, disappearance of the attractor, instant transition to chaos, or cascades of period-adding.
For each DIB of limit cycles studied in the review the authors present the discontinuity mapping, the map that captures the effect of the presence of the boundary on the return map. Also each type of DIB is discussed with a canonical example and an example arising in a practical applications such as friction oscillators, impact oscillators, DC-DC converters, and problems in control theory.

MSC:

34A36 Discontinuous ordinary differential equations
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37N05 Dynamical systems in classical and celestial mechanics
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
34C23 Bifurcation theory for ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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