Stochastically perturbed sliding motion in piecewise-smooth systems
David J. W. Simpson,Rachel Kuske +1 more
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TLDR
In this paper, the effects of small-amplitude, additive, white Gaussian noise on stable sliding motion were quantitatively studied and the mean and variance for the near sliding solution were calculated.Abstract:
Sliding motion is evolution on a switching manifold of a
discontinuous, piecewise-smooth system of ordinary differential equations.
In this paper we quantitatively study the effects of
small-amplitude, additive, white Gaussian noise on stable sliding motion.
For equations that are static in directions parallel to the switching manifold,
the distance of orbits from the switching manifold approaches a quasi-steady-state density.
From this density we calculate the mean and variance for the near sliding solution.
Numerical results of a relay control system
reveal that the noise may significantly affect the period
and amplitude of periodic solutions with sliding segments.read more
Citations
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Hidden dynamics in models of discontinuity and switching
TL;DR: In this paper, the authors generalize Filippov's method to nonlinear sliding modes and show that the nonlinear terms persist in more precise models, for example when the discontinuity is smoothed out.
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Dynamics at a switching intersection: hierarchy, isonomy, and multiple-sliding
TL;DR: The standard ‘Filippov’ method is extended to intersections of discontinuity manifolds in the most natural way possible, allowing more general systems than typical linear control forms to be solved.
Journal ArticleDOI
Metastability for discontinuous dynamical systems under Lévy noise: Case study on Amazonian Vegetation.
Larissa Serdukova,Larissa Serdukova,Yayun Zheng,Jinqiao Duan,Jinqiao Duan,Jürgen Kurths,Jürgen Kurths,Jürgen Kurths +7 more
TL;DR: It is concluded that even a very slight threat to the forest state stability represents L´evy noise with large jumps of low intensity, that can be interpreted as a fire occurring in a non-drought year.
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The Ghosts of Departed Quantities in Switches and Transitions
TL;DR: In this article, the authors discuss the way transitions can be reduced to discontinuities without trivializing them, by preserving so-called hidden terms, and present a prototype for piecewise-smooth models from the asymptotics of systems with rapid transitions.
References
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Brownian Motion and Stochastic Calculus
TL;DR: In this paper, the authors present a characterization of continuous local martingales with respect to Brownian motion in terms of Markov properties, including the strong Markov property, and a generalized version of the Ito rule.
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TL;DR: The kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics, algebraic geometry interacts with physics, and such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes.
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Carl M. Bender,Steven A. Orszag +1 more
TL;DR: A self-contained presentation of the methods of asymptotics and perturbation theory, methods useful for obtaining approximate analytical solutions to differential and difference equations is given in this paper.
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Stochastic differential equations : an introduction with applications
TL;DR: Some Mathematical Preliminaries as mentioned in this paper include the Ito Integrals, Ito Formula and the Martingale Representation Theorem, and Stochastic Differential Equations.