Piecewise smooth dynamical systems: Persistence of periodic solutions and normal forms

Gouveia, Márcio Ricardo Alves; Llibre, Jaume; Novaes, Douglas D.; Pessoa, Claudio

doi:10.1016/j.jde.2015.12.034

Cited by 13 publications

(5 citation statements)

References 22 publications

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“…In Section 3.1, we present a change of coordinates so that system (13) reads in the standard form (3) to apply the averaging method. In Section 3.2, we construct the averaging functions f 1 and f 2 for system (13), defined in (11). Finally, in Section 3.3 we present some trigonometric relations that will be used in the calculus of the zeros of the functions f 1 and f 2 .…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…, where g 1 is given in (22) for 0 ≤ ≤ m. Now, we compute the bifurcation function f 2 defined also in (11).…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…The first studies on this subject considered smooth differential systems and, since then, many contributions have been made in this direction (see [15] and the references therein). The study of limit cycles has also been considered for continuous (see, for instance, [4,23,25]) and discontinuous piecewise smooth differential systems (see, for instance, [11,14,19,20,26]). Most of them are concentrated on planar piecewise differential systems.…”

Section: Introductionmentioning

confidence: 99%

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Limit cycles of piecewise polynomial perturbations of higher dimensional linear differential systems

2019

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The averaging theory has been extensively employed for studying periodic solutions of smooth and nonsmooth differential systems. Here, we extend the averaging theory for studying periodic solutions a class of regularly perturbed non-autonomous n-dimensional discontinuous piecewise smooth differential system. As a fundamental hypothesis, it is assumed that the unperturbed system has a manifold Z ⊂ R n of periodic solutions satisfying dim(Z) < n. Then, we apply this result to study limit cycles bifurcating from periodic solutions of linear differential systems, x = M x, when they are perturbed inside a class of discontinuous piecewise polynomial differential systems with two zones. More precisely, we study the periodic solutions of the following differential system x = M x + εF n 1 (x) + ε 2 F n 2 (x), in R d+2 where ε is a small parameter, M is a (d+2)×(d+2) matrix having one pair of pure imaginary conjugate eigenvalues, m zeros eigenvalues, and d − m non-zero real eigenvalues.

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Section: Preliminary Resultsmentioning

confidence: 99%

“…, where g 1 is given in (22) for 0 ≤ ≤ m. Now, we compute the bifurcation function f 2 defined also in (11).…”

Section: Preliminary Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Limit cycles of piecewise polynomial perturbations of higher dimensional linear differential systems

2019

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“…[34,97]), to describe bifurcations in an orbit's intersection with the discontinuity one has the discontinuity mappings (see [34,110]), and to study the separation of orbits to establish connections between manifolds or existence of limit cycles one may extend the idea of Melnikov functions (see e.g. [12,62,91], though at present there are large and increasing number of papers investigating this line of study). There is also work ongoing to extend other powerful tools, such as those of inverse integrating factors [23].…”

Section: The Planmentioning

confidence: 99%

Modeling with Nonsmooth Dynamics

Jeffrey

2020

Frontiers in Applied Dynamical Systems: Reviews and Tutorials

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As mathematics is applied to model ever new problems in engineering and the life sciences, increasing use is being made of systems that switch between different sets of equations on distinct domains. To find their dynamics requires the discontinuity between domains to be resolved or 'regularized' in some way, and there exist a range of methods to do so. Some preserve the ideal character of the discontinuity as a piecewise-smooth system (giving e.g. 'impact' or 'switching' dynamics), while others blur the discontinuity by smoothing it out, or introducing overshoots due to deterministic or stochastic delays.Despite exciting new applications and major theoretical advances, it remains unclear how widely applicable nonsmooth models are, or in what sense they approximate discontinuities in real world systems. It is even unclear how to correctly simulate or solve nonsmooth systems, or how robust such solutions are to perturbation. To move closer towards these goals, here we survey one of the main approaches to modeling nonsmooth dynamics, and look at how loosening some of its rigourous but idealized framework allows us to probe its modeling assumptions. We also draw together a range of phenomena that characterize the sensitivity and robustness of nonsmooth dynamical models.

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“…Some works addressing three-dimensional Filippov systems are [3], [4], [5], [7], [9], [11], [12], [14], [18] and [19]. In larger dimensions, the papers [1] and [16] present the existence and uniqueness of a specific sliding vector field on a discontinuity surface of co-dimension 2.…”

mentioning

confidence: 99%

Limit sets near a cusp-fold singularity in a class of 3D non-smooth vector fields

Euzébio,

Romero

2024

Preprint

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We consider a class of three-dimensional non-smooth vector fields presenting a {\it cusp-fold singularity}. We develop a local approach to classify the limit sets of points positioned near the cusp-fold. The results state the existence of center manifolds connecting to the cusp-fold, the absence of isolated periodic orbits of any kind and sufficient conditions for the asymptotic stability of the cusp-fold. The results bring new, relevant material to the subject of structural stability and bifurcations of co-dimension one in three-dimensional non-smooth vector fields. Mathematics Subject Classification (2020) MSC 34A36 · 34C05 · 37C27

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Piecewise smooth dynamical systems: Persistence of periodic solutions and normal forms

Cited by 13 publications

References 22 publications

Limit cycles of piecewise polynomial perturbations of higher dimensional linear differential systems

Limit cycles of piecewise polynomial perturbations of higher dimensional linear differential systems

Modeling with Nonsmooth Dynamics

Limit sets near a cusp-fold singularity in a class of 3D non-smooth vector fields

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