Sliding Vector Fields via Slow--Fast Systems

Llibre, Jaume; Silva, Paulo Ricardo da; Teixeira, Marco Antônio

doi:10.36045/bbms/1228486412

Cited by 44 publications

(56 citation statements)

References 12 publications

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“…Proposition 5.1 also motivates the conditions (9) for the function φ(y), as explained in the following Remark. that is a graph ofŷ over Σ s,Filippov [25,30]. In terms of (x, y, θ) this set collapses onto Σ s,Filippov instead of Σ s , and hence Z ST ε (z) does not tend to Z(z) as ε → 0.…”

Section: Regularizationmentioning

confidence: 99%

Canards in Stiction: On Solutions of a Friction Oscillator by Regularization

Bossolini¹,

Brøns²,

Kristiansen³

2017

SIAM J. Appl. Dyn. Syst.

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We study the solutions of a friction oscillator subject to stiction. This discontinuous model is non-Filippov, and the concept of Filippov solution cannot be used. Furthermore some Carathéodory solutions are unphysical. Therefore we introduce the concept of stiction solutions: these are the Carathéodory solutions that are physically relevant, i.e. the ones that follow the stiction law. However, we find that some of the stiction solutions are forward non-unique in subregions of the slip onset. We call these solutions singular, in contrast to the regular stiction solutions that are forward unique. In order to further the understanding of the non-unique dynamics, we introduce a regularization of the model. This gives a singularly perturbed problem that captures the main features of the original discontinuous problem. We identify a repelling slow manifold that separates the forward slipping to forward sticking solutions, leading to a high sensitivity to the initial conditions. On this slow manifold we find canard trajectories, that have the physical interpretation of delaying the slip onset. We show with numerics that the regularized problem has a family of periodic orbits interacting with the canards. We observe that this family has a saddle stability and that it connects, in the rigid body limit, the two regular, slip-stick branches of the discontinuous problem, that were otherwise disconnected.

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Section: Regularizationmentioning

confidence: 99%

Canards in Stiction: On Solutions of a Friction Oscillator by Regularization

Bossolini¹,

Brøns²,

Kristiansen³

2017

SIAM J. Appl. Dyn. Syst.

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“…RELAXED FILIPPOV SYSTEMS Teixeira has developed a framework for approximating the dynamics of autonomous, piecewise-smooth vector fields using a parameterized family of smooth, stiff vector fields (see e.g. [9]). Here, we generalize the approach to control systems of the form (1).…”

Section: Filippov Solutionsmentioning

confidence: 99%

“…However, numerical tools for simulating, analyzing and controlling Filippov systems are far less developed than they are for smooth dynamical systems. Therefore, we build on the work of [9] and demonstrate how to approximate the hybrid Filippov solution using a parameterized family of smooth, stiff control systems defined on the relaxed topology T. Westenbroek from [10]. Under standard regularity assumptions, these relaxations are shown to recover the hybrid Filippov solution as the appropriate limit is taken.…”

Section: Introductionmentioning

confidence: 99%

A New Solution Concept and Family of Relaxations for Hybrid Dynamical Systems

Westenbroek

González²,

Sastry

2018

2018 IEEE Conference on Decision and Control (CDC)

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We introduce a holistic framework for the analysis, approximation and control of the trajectories of hybrid dynamical systems which display event-triggered discrete jumps in the continuous state. We begin by demonstrating how to explicitly represent the dynamics of this class of systems using a single piecewise-smooth vector field defined on a manifold, and then employ Filippov's solution concept to describe the trajectories of the system. The resulting hybrid Filippov solutions greatly simplify the mathematical description of hybrid executions, providing a unifying solution concept with which to work. Extending previous efforts to regularize piecewisesmooth vector fields, we then introduce a parameterized family of smooth control systems whose trajectories are used to approximate the hybrid Filippov solution numerically. The two solution concepts are shown to agree in the limit, under mild regularity conditions.

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“…For ǫ = 0, let S be the set of all singular points of (4). We call S the slow manifold of the singular perturbation problem and it is important to notice that equation (5) defines a dynamical system, on S, called the reduced problem: (7) f (x, y, 0) = 0,ẏ = g(x, y, 0).…”

Section: Canard Cycles and Singular Perturbations Problemsmentioning

confidence: 99%

Closed poly-trajectories and Poincaré index of non-smooth vector fields on the plane

2013

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This paper is concerned with closed orbits of non-smooth vector fields on the plane. For a subclass of non-smooth vector fields we provide necessary and sufficient conditions for the existence of canard kind solutions. By means of a regularization we prove that the canard cycles are singular orbits of singular perturbation problems which are limit periodic sets of a sequence of limit cycles. Moreover, we generalize the Poincaré Index for non-smooth vector fields.1991 Mathematics Subject Classification. Primary 34C20, 34C26, 34D15, 34H05.

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Sliding Vector Fields via Slow--Fast Systems

Cited by 44 publications

References 12 publications

Canards in Stiction: On Solutions of a Friction Oscillator by Regularization

Canards in Stiction: On Solutions of a Friction Oscillator by Regularization

A New Solution Concept and Family of Relaxations for Hybrid Dynamical Systems

Closed poly-trajectories and Poincaré index of non-smooth vector fields on the plane

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