Phase portraits of continuous piecewise linear Liénard differential systems with three zones

Li, Shimin; Llibre, Jaume

doi:10.1016/j.chaos.2018.12.037

Cited by 7 publications

(3 citation statements)

References 24 publications

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“…Proof. Note that for any regularization function ρ, defining ρ as in Proposition 4, the vectors fields F ρ and F ρ are topologically equivalent since they are proportional by a positive function, see ( 7)- (8). Moreover, the topological equivalence is the identity.…”

Section: For Short Let Us Define the Functionmentioning

confidence: 99%

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Poincaré compactification for non-polynomial vector fields

Bravo¹,

Fernández²,

Teruel³

2020

Preprint

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In this work a theorical framework to apply the Poincaré compactification technique to locally Lipschitz continuous vector fields is developed. It is proved that these vectors fields are compactifiable in the n-dimensional sphere, though the compactified vector field can be identically null in the equator. Moreover, for a fixed projection to the hemisphere, all the compactifications of a vector field, which are not identically null on the equator are equivalent. Also, the conditions determining the invariance of the equator for the compactified vector field are obtained. Up to the knowledge of the authors, this is the first time that the Poincaré compactification of locally Lipschitz continuous vector fields is studied.These results are illustrated applying them to some families of vector fields, like polynomial vector fields, vector fields defined as a sum of homogeneous functions and vector fields defined by piecewise linear functions.

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Section: For Short Let Us Define the Functionmentioning

confidence: 99%

“…Therefore, the piecewise linear function ϕ(σ) defined as The compactification of the continuous PWL vector fields has been also addressed in some papers. Nevertheless, every time just for a particular group of these vector fields [8,9]. Next we consider the general case.…”

Section: Families Of Non-null Poincaré Compactifiable Vector Fielsmentioning

confidence: 99%

Poincaré compactification for non-polynomial vector fields

Bravo¹,

Fernández²,

Teruel³

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…In fact, as far as we know, the studies found in the pertinent literature deal with piecewise smooth systems with three zones in the plane with at least one equilibrium point which can be real or virtual. This is the case, for instance, in the studies carried out in [3], [6], [7], [8], [9], [10]. We emphasize that in these articles the authors studied the existence and the number of limit cycles under the hypothesis of the continuity of the vector fields in the separation curves which are straight lines.…”

mentioning

confidence: 94%