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.

This paper describes how to use smooth solvers for simulation of a class of piecewise smooth dynamical systems, called Filippov systems, with discontinuous vector fields. In these systems constrained motion along a discontinuity surface (so-called sliding) is possible and require special treatment numerically. The introduced algorithms are based on an extension to Filippov's method to stabilize the sliding flow together with accurate detection of the entrance and exit of sliding regions. The methods are implemented in a general way in Matlab and sufficient details are given to enable users to modify the code to run on arbitraray examples. Here, the method is used to compute the dynamics of three example systems, a dry-friction oscillator, a relay feedback system and a model of a oil well drill-string.

This paper proposes a strategy for the classification of codimension-two discontinuity-induced bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (also known as C-bifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a nongeneric way, such as grazing contact. Several such codimension-one events have recently been identified, causing for example, period-adding or sudden onset of chaos. Here, the focus is on codimension-two grazings that are local in the sense that the dynamics can be fully described by an appropriate Poincaré map from a neighborhood of the grazing point (or points) of the critical cycle to itself. It is proposed that codimension-two grazing bifurcations can be divided into three distinct types: either the grazing point is degenerate, or the grazing cycle is itself degenerate (e.g. nonhyperbolic) or we have the simultaneous occurrence of two grazing events. A careful distinction is drawn between their occurrence in systems with discontinuous states, discontinuous vector fields, or that with discontinuity in some derivative of the vector field. Examples of each kind of bifurcation are presented, mostly derived from mechanical applications. For each example, where possible, principal bifurcation curves characteristic to the codimension-two scenario are presented and general features of the dynamics discussed. Many avenues for future research are opened.

This paper considers dynamical systems that are derived from mechanical systems with impacts. In particular we will focus on chatteringaccumulation of impacts-for which local discontinuity mappings will be derived. We will first show how to use these mappings in simulation schemes, and secondly how the mappings are used to calculate the stability of limit cycles with chattering by solving the first variational equations.

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Within decisions, perceived alternatives compete until one is preferred. Across decisions, the playing field on which these alternatives compete evolves to favor certain alternatives. Mouse cursor trajectories provide rich continuous information related to such cognitive processes during decision making. In three experiments, participants learned to choose symbols to earn points in a discrimination learning paradigm and the cursor trajectories of their responses were recorded. Decisions between two choices that earned equally high-point rewards exhibited far less competition than decisions between choices that earned equally low-point rewards. Using positional coordinates in the trajectories, it was possible to infer a potential field in which the choice locations occupied areas of minimal potential. These decision spaces evolved through the experiments, as participants learned which options to choose. This visualisation approach provides a potential framework for the analysis of local dynamics in decision-making that could help mitigate both theoretical disputes and disparate empirical results.

Simultaneous impacts of multiple objects in mechanical systems are considered. It will be shown that in some cases the dynamics in a neighbourhood of such events can be explained by piecewise linear two-dimensional maps. These maps give rise to a variety of complex dynamics, which will be analysed. A special emphasis will be put on period-adding cascades.