A comparison of Filippov sliding vector fields in codimension 2

Dieci, Luca; Difonzo, Fabio V.

doi:10.1016/j.cam.2013.10.055

Cited by 24 publications

(17 citation statements)

References 9 publications

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“…Others (e.g., [7,16]) have argued in a more general context for using a multilinear interpolation of these vectors, rather than the convex hull, but this approach only corresponds to the concept of Filippov solutions in the narrow sense if the differential equations are multilinear in the Z i . There has been a profusion of names for these alternative definitions of vector field and solution, including Filippov solution in the narrow sense [17,19], Utkin's vector field or solution [7], the Aizermann-Pyatnitskii vector field [1], the bilinear interpolation vector field [6] or the convex canopy [16]. See also [12] for discussion of many of these papers.…”

Section: Continuous-time Switching Networkmentioning

confidence: 99%

“…The fixed point in the switching intersection (found by setting the Z 1 and Z 2 to 0) is at Z * 1 = 1 2 , Z * 2 = 6 10+3α . So, for example, at α = − 1 3 , 0 and 1 3 , we have Z * 2 = 2 3 , 3 5 and 6 11 , respectively. The Jacobian matrix is 1 3 ] (the allowed range), and tr(J) < 0 if and only if α > 0 (within the allowed range).…”

Section: Singular Perturbation Analysismentioning

confidence: 99%

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Zeno breaking, the Contact effect and sensitive behaviour in piecewise-linear systems

Edwards

2018

Eur. J. Appl. Math

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Non-smooth approximations of steep sigmoidal switching networks, such as those used as qualitative models of gene regulation, lead to analytic and computational challenges that arise as a result of the discontinuities in the vector fields. In order to highlight the need for care in dealing with such systems, several particular phenomena are presented here through illustrative examples, including ‘Zeno breaking’, or computing beyond the finite time convergence of an infinite sequence of threshold transitions; the ‘Contact’ effect, in which in the discontinuous limit, trajectories can pass through a ‘saddle point’ without stopping, though these solutions are not unique and other solutions stop for arbitrary time intervals; and sensitive behaviour that arises from exotic dynamics within switching regions.

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Section: Continuous-time Switching Networkmentioning

confidence: 99%

Section: Singular Perturbation Analysismentioning

confidence: 99%

Zeno breaking, the Contact effect and sensitive behaviour in piecewise-linear systems

Edwards

2018

Eur. J. Appl. Math

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“…This approach appears in [26] for a surface of co-dimension 1. For the case of Σ of co-dimension 2, it has been studied first in [2] and then in [8], always in the case of nodally attractive Σ. The idea here is that one has a region U ǫ around Σ (called a chatterbox in [2]), and uses the same vector field, say f 1 , not only in the region R 1 , but until the boundary of U ǫ in a different region is reached; at that point, a switch to the appropriate vector field in the new region is performed.…”

Section: Introductionmentioning

confidence: 99%

Piecewise smooth systems near a co-dimension 2 discontinuity manifold: Can one say what should happen?

Dieci¹,

Elia²

2016

Discrete &Amp; Continuous Dynamical Systems - S

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We consider a piecewise smooth system in the neighborhood of a co-dimension 2 discontinuity manifold Σ (intersection of two co-dimension 1 manifolds). Within the class of Filippov solutions, if Σ is attractive, one should expect solution trajectories to slide on Σ. It is well known, however, that the classical Filippov convexification methodology does not render a uniquely defined sliding vector field on Σ. The situation is further complicated by the possibility that, regardless of how sliding on Σ is taking place, during sliding motion a trajectory encounters so-called generic first order exit points, where Σ ceases to be attractive.In this work, we attempt to understand what behavior one should expect of a solution trajectory near Σ when Σ is attractive, what to expect when Σ ceases to be attractive (at least, at generic exit points), and finally we also contrast and compare the behavior of some regularizations proposed in the literature, whereby the original piecewise smooth system is replaced -in a neighborhood of Σ-by a smooth differential system.Through analysis and experiments in R 3 and R 4 , we will confirm some known facts, and provide some important insight: (i) when Σ is attractive, a solution trajectory indeed does remain near Σ, viz. sliding on Σ is an appropriate idealization (of course, in general, one cannot predict which sliding vector field should be selected); (ii) when Σ loses attractivity (at first order exit conditions), a typical solution trajectory leaves a neighborhood of Σ; (iii) there is no obvious way to regularize the system so that the regularized trajectory will remain near Σ as long as Σ is attractive, and so that it will be leaving (a neighborhood of) Σ when Σ looses attractivity.We reach the above conclusions by considering exclusively the given piecewise smooth system, without superimposing any assumption on what kind of dynamics near Σ (or sliding motion on Σ) should have been taking place. The only datum for us is the original piecewise smooth system, and the dynamics inherited by it.1991 Mathematics Subject Classification. 34A36, 65P99.

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“…The modeling and control of the ecological systems and the evolution of species are known to undergo a variety of bifurcation phenomena ranging from multiplicity and stability of steady states to sustained oscillations. Recently, Filippov systems have gained considerable attention in life science and engineering [3,10,11,12,16,20,21,24,31,32,34] and they provide a natural and convenient unified framework for mathematical modeling of several real-world problems. In this paper, we proposed a Filippov food chain model with food chain control strategy to describe three species food chain interaction model composed of x(t), y(t), and z(t).…”

mentioning

confidence: 99%

Stability and bifurcation analysis of Filippov food chain system with food chain control strategy

Hamdallah¹,

Tang²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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In the present work, we introduce a control model to describe three species food chain interaction model composed of prey, middle predator, and top predator. The middle predator preys on prey and the top predator preys on middle predator. The control techniques of the exploited natural resources are used to modulate the harvesting effort to avoid high risks of extinction of the middle predator and keep stability of the food chain, by prohibiting fishing when the population density drops below a prescribed threshold. The behavior of the system stability of the regular, virtual, pseudo-equilibrium and tangent points are discussed. The complicated non-smooth dynamic behaviors (sliding and crossing segment and their domains) are analyzed. The bifurcation set of pseudo-equilibrium and the sliding crossing bifurcation have been investigated. Our analytical findings are verified through numerical investigations.

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A comparison of Filippov sliding vector fields in codimension 2

Cited by 24 publications

References 9 publications

Zeno breaking, the Contact effect and sensitive behaviour in piecewise-linear systems

Zeno breaking, the Contact effect and sensitive behaviour in piecewise-linear systems

Piecewise smooth systems near a co-dimension 2 discontinuity manifold: Can one say what should happen?

Stability and bifurcation analysis of Filippov food chain system with food chain control strategy

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