Perturbation Theory for Non-smooth Systems

Teixeira, Marco Antônio

doi:10.1007/978-0-387-30440-3_400

Cited by 44 publications

(40 citation statements)

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“…Observe that A h (θ) is rather technical but it plays an important role in the implementation of Theorem 1 for system (12). Note also that it depends only on function h.…”

Section: (14)mentioning

confidence: 99%

“…Proof of Fundamental Lemma: Consider system (12) and assume that for this system we verify A h (θ) = 0 for all θ ∈ [0, 2π). Since the periodic solutions of system (1), that we are perturbing, live on the cylinder C, we will perform a cylindrical change of coordinates in system (12) by introducing the new variables (z, r, θ) given implicitly by x = r cos θ, y = r sin θ and z = z.…”

Section: Proofs and Examplesmentioning

confidence: 99%

“…Non-smooth vector fields have become certainly one of the common frontiers between Mathematics and Physics or Engineering. Many authors have contributed to the study of non-smooth vector fields (see for instance the pioneering work [7] or the didactic works [3,12], and references therein about details of these multi-valued vector fields). In our approach Filippov's convention is considered, see [7].…”

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confidence: 99%

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Bifurcation of limit cycles from a non-smooth perturbation of a two-dimensional isochronous cylinder

Buzzi

Euzébio

Mereu

2016

Bulletin des Sciences Mathématiques

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Abstract. Detect the birth of limit cycles in non-smooth vector fields is a very important matter into the recent theory of dynamical systems and applied sciences. The goal of this paper is to study the bifurcation of limit cycles from a continuum of periodic orbits filling up a two-dimensional isochronous cylinder of a vector field in R 3 . The approach involves the regularization process of non-smooth vector fields and a method based in the Malkin's bifurcation function for C 0 perturbations. The results provide sufficient conditions in order to obtain limit cycles emerging from the cylinder through smooth and non-smooth perturbations of it. To the best of our knowledge they also illustrate the implementation by the first time of a new method based in the Malkin's bifurcation function. In addition, some points concerning the number of limit cycles bifurcating from non-smooth perturbations compared with smooth ones are studied. In summary the results yield a better knowledge about limit cycles in non-smooth vector fields in R 3 and explicit a manner to obtain them by performing non-smooth perturbations in codimension one Euclidean manifolds.

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“…Observe that A h (θ) is rather technical but it plays an important role in the implementation of Theorem 1 for system (12). Note also that it depends only on function h.…”

Section: (14)mentioning

confidence: 99%

Section: Proofs and Examplesmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Bifurcation of limit cycles from a non-smooth perturbation of a two-dimensional isochronous cylinder

Buzzi

Euzébio

Mereu

2016

Bulletin des Sciences Mathématiques

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show abstract

“…The theory of piecewise smooth vector fields (PSVF for short) has been developing very fast in the last years, mainly due to its strong relation with branches of applied science, such as mechanical, aerospace engineering, electrical and electronic engineering, physics, economics, among others areas. Indeed, PSVF are in the boundary between mathematics, physics and engineering, see [19] and [23] for a recent survey on this subject.…”

Section: Introductionmentioning

confidence: 99%

“…One of the biggest limitations of the theory of PSVF is the lack of global results, see [23]. In [3], authors considered conditions to get structural stability for PSVF defined in a compact differentiable manifolds, like a version of the Peixoto's Theorem.…”

Section: Introductionmentioning

confidence: 99%