824 research outputs found
Hidden dynamics in models of discontinuity and switching
AbstractSharp switches in behaviour, like impacts, stickāslip motion, or electrical relays, can be modelled by differential equations with discontinuities. A discontinuity approximates fine details of a switching process that lie beyond a bulk empirical model. The theory of piecewise-smooth dynamics describes what happens assuming we can solve the system of equations across its discontinuity. What this typically neglects is that effects which are vanishingly small outside the discontinuity can have an arbitrarily large effect at the discontinuity itself. Here we show that such behaviour can be incorporated within the standard theory through nonlinear terms, and these introduce multiple sliding modes. We show that the nonlinear terms persist in more precise models, for example when the discontinuity is smoothed out. The nonlinear sliding can be eliminated, however, if the model contains an irremovable level of unknown error, which provides a criterion for systems to obey the standard Filippov laws for sliding dynamics at a discontinuity
Non-deterministic dynamics of a mechanical system
A mechanical system is presented exhibiting a non-deterministic singularity,
that is, a point in an otherwise deterministic system where forward time
trajectories become non-unique. A Coulomb friction force applies linear and
angular forces to a wheel mounted on a turntable. In certain configurations the
friction force is not uniquely determined. When the dynamics evolves past the
singularity and the mechanism slips, the future state becomes uncertain up to a
set of possible values. For certain parameters the system repeatedly returns to
the singularity, giving recurrent yet unpredictable behaviour that constitutes
non-deterministic chaotic dynamics. The robustness of the phenomenon is such
that we expect it to persist with more sophisticated friction models,
manifesting as extreme sensitivity to initial conditions, and complex global
dynamics attributable to a local loss of determinism in the limit of
discontinuous friction.Comment: 22 pages, 8 figure
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