Sliding Motion in Filippov Differential Systems: Theoretical Results and a Computational Approach

Dieci, Luca; Lopez, L.

doi:10.1137/080724599

Cited by 87 publications

(80 citation statements)

References 30 publications

(63 reference statements)

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“…There is an ambiguity, because three parameters among the λ j have to be determined to satisfy the two conditions α 1 (y)ẏ = 0 and α 2 (y)ẏ = 0, and much research is devoted to this question, see [2,1].…”

Section: Connection With Filippov Solutionsmentioning

confidence: 99%

Regularization of Neutral Delay Differential Equations with Several Delays

Guglielmi

Hairer

2013

J Dyn Diff Equat

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Summary For neutral delay differential equations the right-hand side can be multi-valued, when one or several delayed arguments cross a breaking point. This article studies a regularization via a singularly perturbed problem, which smooths the vector field and removes the discontinuities in the derivative of the solution. A low-dimensional dynamical system is presented, which characterizes the kind of generalized solution that is approximated. For the case that the solution of the regularized problem has high frequency oscillations around a codimension-2 weak solution of the original problem, a new stabilizing regularization is proposed and analyzed.Mathematics Subject Classification (2010): 34K40, 34K26, 34E05

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Section: Connection With Filippov Solutionsmentioning

confidence: 99%

Regularization of Neutral Delay Differential Equations with Several Delays

Guglielmi

Hairer

2013

J Dyn Diff Equat

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“…However, in general, there is no uniquely defined Filippov sliding vector. For the case of general value of p, under appropriate attractivity assumptions (the "nodal attractivity" conditions of the present work), in [16] we proposed a systematic approach to select an unique, well defined, sliding vector field. However, the choice we made in [16] does not necessarily reduce to a particular choice of a Filippov vector field of the form in (1.5), and this fact prompted us to reconsider the issue in this work.…”

Section: Fig 1 Regions and Surfacesmentioning

confidence: 99%

“…There are two main advantages of selecting a sliding vector field (1.5), rather than dealing with the more complicated differential inclusion (1.4): (a) it is simple to develop numerical methods during the sliding regime (e.g., as done in [16]), and (b) it becomes possible to carry out a non-ambiguous study of the dynamics of the system (e.g., see Examples 5.2, 5.3, 5.4).…”

Section: Fig 1 Regions and Surfacesmentioning

confidence: 99%

“…Recently, two different approaches have been proposed to eliminate this ambiguity when satisfies appropriate attractivity assumptions (the nodal attractivity assumptions below). The first approach is due to Alexander and Seidman and is based on sigmoid blending techniques (see [2,14]), the second one is based on geometric considerations and we proposed it in [16].…”

Section: Co-dimension Casementioning

confidence: 99%

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Sliding motion on discontinuity surfaces of high co-dimension. A construction for selecting a Filippov vector field

Dieci

Lopez

2011

Numer. Math.

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In this paper we consider the issue of sliding motion in Filippov systems on the intersection of two or more surfaces. To this end, we propose an extension of the Filippov sliding vector field on manifolds of co-dimension p, with p ≥ 2. Our model passes through the use of a multivalued sign function reformulation. To justify our proposal, we will restrict to cases where the sliding manifold is attractive. For the case of co-dimension p = 2, we will distinguish between two types of attractive sliding manifold: "node-like" and "spiral-like". The case of node-like attractive manifold will be further extended to the case of p ≥ 3. Finally, we compare our model to other existing methodologies on some examples.

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“…Classical qualitative theory based on smooth dynamical systems cannot satisfactorily explain phenomena such as switching and hysteresis in electronic circuits, saturation effects in control systems or friction and impacting behaviors in mechanical systems [2][3][4]. Therefore, PWS systems of ordinary differential equations are being used to model in a more realistic form these inherent nonsmooth phenomena [5].…”

Section: Introductionmentioning

confidence: 99%

Topological Classification of Limit Cycles of Piecewise Smooth Dynamical Systems and Its Associated Non-Standard Bifurcations

Taborda

Arango

2014

Entropy

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Abstract:In this paper, we propose a novel strategy for the synthesis and the classification of nonsmooth limit cycles and its bifurcations (named Non-Standard Bifurcations or Discontinuity Induced Bifurcations or DIBs) in n-dimensional piecewise-smooth dynamical systems, particularly Continuous PWS and Discontinuous PWS (or Filippov-type PWS) systems. The proposed qualitative approach explicitly includes two main aspects: multiple discontinuity boundaries (DBs) in the phase space and multiple intersections between DBs (or corner manifolds-CMs). Previous classifications of DIBs of limit cycles have been restricted to generic cases with a single DB or a single CM. We use the definition of piecewise topological equivalence in order to synthesize all possibilities of nonsmooth limit cycles. Families, groups and subgroups of cycles are defined depending on smoothness zones and discontinuity boundaries (DB) involved. The synthesized cycles are used to define bifurcation patterns when the system is perturbed with parametric changes. Four families of DIBs of limit cycles are defined depending on the properties of the cycles involved. Well-known and novel bifurcations can be classified using this approach.

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Sliding Motion in Filippov Differential Systems: Theoretical Results and a Computational Approach

Cited by 87 publications

References 30 publications

Regularization of Neutral Delay Differential Equations with Several Delays

Regularization of Neutral Delay Differential Equations with Several Delays

Sliding motion on discontinuity surfaces of high co-dimension. A construction for selecting a Filippov vector field

Topological Classification of Limit Cycles of Piecewise Smooth Dynamical Systems and Its Associated Non-Standard Bifurcations

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