Sliding Shilnikov connection in Filippov-type predator–prey model

Carvalho, Tiago; Novaes, Douglas D.; Gonçalves, Luiz Fernando

doi:10.1007/s11071-020-05672-w

Cited by 20 publications

(7 citation statements)

References 35 publications

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“…where = 𝑑 /𝑑𝑡, x = x(𝑡) and y = y(𝑡). As 0 < 𝜉 < 1, then (9) and (10) have exactly the same phase portrait, except for the trajectories speed, which is greater for first system and smaller for the second. Therefore, the following definition makes sense: Definition 2.…”

Section: Geometrical Singular Perturbation Theorymentioning

confidence: 87%

“…Therefore, the following definition makes sense: Definition 2. 3 We say that ( 9) and ( 10) form a ( , )-slowfast system with fast system given by (9) and slow system given by (10).…”

Section: Geometrical Singular Perturbation Theorymentioning

confidence: 99%

“…Generally, we write F = (F + , F − ) to denote the elements of this set. These systems arise frequently, for instance, in the study of mechanical systems with impact or friction [6,25,31,50], electronic circuits with switches [4,5,10,47], biological and climate models with abrupt changes [3,9,32,37,38], economics and politics [1,45,46], etc. Hence, not only due to its applications, but also, its mathematical beauty, the theory of non-smooth dynamical system is a very active field, attracting and mobilizing scientists from all around the world.…”

Section: Introductionmentioning

confidence: 99%

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Dynamics and Stability of Non-Smooth Dynamical Systems with Two Switches

Silva

Martins

2021

Preprint

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One of the most common hypotheses on the theory of non-smooth dynamical systems is a regular surface as switching manifold, at which case there is at least well-defined and established Filippov dynamics. However, systems with singular switching manifolds still lack such well-established dynamics, although present in many relevant models of phenomena where multiple switches or multiple abrupt changes occur. At this work, we leverage a methodology that, through blow-ups and singular perturbation, allows the extension of Filippov dynamics to the singular case. Specifically, tridimensional systems whose switching manifold consists of an algebraic manifold with transversal self-intersection are considered. This configuration, known as double discontinuity, represents systems with two switches and whose singular part consists of a straight line, where ordinary Filippov dynamics is not directly applicable. For the general, non-linear case, beyond defining the so-called fundamental dynamics over the singular part, general theorems on its qualitative behavior are provided. For the affine case, however, theorems fully describing the fundamental dynamics are obtained. Finally, this fine-grained control over the dynamics is leveraged to derive Peixoto-like theorems characterizing semi-local structural stability.

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Section: Geometrical Singular Perturbation Theorymentioning

confidence: 87%

“…Therefore, the following definition makes sense: Definition 2. 3 We say that ( 9) and ( 10) form a ( , )-slowfast system with fast system given by (9) and slow system given by (10).…”

Section: Geometrical Singular Perturbation Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dynamics and Stability of Non-Smooth Dynamical Systems with Two Switches

Silva

Martins

2021

Preprint

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“…A non-exhaustive list of books concerning this theme includes [1,6,9,15,19,24]. Also, papers like [2,3,5,10,11,13,17,22,23] deals with applications of such theory to real world phenomena. Without doubts, the most valuable text concerning PSVFs is the book [9] of Filippov.…”

Section: Introductionmentioning

confidence: 99%

Sliding motion on tangential sets of Filippov systems

Carvalho¹,

Novaes²,

Tonon³

2021

Preprint

Self Cite

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We consider piecewise smooth vector fields Z = (Z + , Z − ) defined in R n where both vector fields are tangent to the switching manifold Σ along a manifold M . Our main purpose is to study the existence of an invariant vector field defined on M , that we call tangential sliding vector field. We provide necessary and sufficient conditions under Z to characterize the existence of this vector field. Considering a regularization process, we proved that this tangential sliding vector field is topological equivalent to an invariant dynamics under the slow manifold. The results are applied to study a Filippov model for intermittent treatment of HIV.

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“…The exact instant when we change the laws of the system is characterized by a switching manifold on the state space where a new vector field is defined as an average between both vector fields. For instance, see [26] for applications in control theory, [12,19] in mechanics models, [6,17] in electrical circuits, [11,16] in relay systems, [7,18,24,25] in biological models, among others.…”

mentioning

confidence: 99%

A flow on $ S^2 $ presenting the ball as its minimal set

Carvalho¹,

Gonçalves²

2021

DCDS-B

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The main goal of this paper is to present the existence of a vector field tangent to the unit sphere S 2 such that S 2 itself is a minimal set. This is reached using a piecewise smooth (discontinuous) vector field and following the Filippov's convention on the switching manifold. As a consequence, none regularization process applied to the initial model can be topologically equivalent to it and we obtain a vector field tangent to S 2 without equilibria.

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Sliding Shilnikov connection in Filippov-type predator–prey model

Cited by 20 publications

References 35 publications

Dynamics and Stability of Non-Smooth Dynamical Systems with Two Switches

Dynamics and Stability of Non-Smooth Dynamical Systems with Two Switches

Sliding motion on tangential sets of Filippov systems

A flow on $ S^2 $ presenting the ball as its minimal set

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