| 1. |
- di Bernardo, M., et al.
(author)
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Bifurcations in Nonsmooth Dynamical Systems
- 2008
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In: SIAM Review. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1445 .- 1095-7200. ; 50:4, s. 629-701
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Research review (peer-reviewed)abstract
- A review is presented of the one-parameter, nonsmooth bifurcations that occur in a variety of continuous-time piecewise-smooth dynamical systems. Motivated by applications, a pragmatic approach is taken to defining a discontinuity-induced bifurcation (DIB) as a nontrivial interaction of a limit set with respect to a codimension-one discontinuity boundary in phase space. Only DIBs that are local are considered, that is, bifurcations involving equilibria or a single point of boundary interaction along a limit cycle for flows. Three classes of systems are considered, involving either state jumps, jumps in the vector field, or jumps in some derivative of the vector field. A rich array of dynamics are revealed, involving the sudden creation or disappearance of attractors, jumps to chaos, bifurcation diagrams with sharp corners, and cascades of period adding. For each kind of bifurcation identified, where possible, a kind of "normal form" or discontinuity mapping (DM) is given, together with a canonical example and an application. The goal is always to explain dynamics that may be observed in simulations of systems which include friction oscillators, impact oscillators, DC-DC converters, and problems in control theory.
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| 2. |
- Edström, Per
(author)
-
A Fast and Stable Solution Method for the Radiative Transfer Problem
- 2005
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In: SIAM Review. - 0036-1445 .- 1095-7200. ; 47:3, s. 447-468
-
Journal article (peer-reviewed)abstract
- Radiative transfer theory considers radiation in turbid media, and is used in a wide range of applications. This paper outlines a problem formulation and a solution method for the radiative transfer problem in multilayer scattering and absorbing media, using discrete ordinate model geometry. A selection of different steps is brought together. The main contribution here is the synthesis of these steps, all of which have been used in different areas, but never all together in one method. First all necessary steps to get a numerically stable solution procedure are treated, and then methods are introduced to increase the speed by a factor of several thousand. This includes methods for handling strongly forward-scattering media. The method is shown to be unconditionally stable, whilst the problem was previously considered numerically intractable.
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