Global phase portraits of the generalized van der Pol systems

Jaume Llibre; Claudia Valls

doi:10.1016/j.bulsci.2022.103213

Global phase portraits of the generalized van der Pol systems

Jaume Llibre, Claudia Valls^*

^*Corresponding author for this work

Department of Mathematics

University of Lisbon

Research output: Contribution to journal › Article › Research › peer-review

2 Citations (Scopus)

Abstract

We consider the generalized van der Pol systems x˙=y,y˙=−x+(1−x²)f(y), where f∈R[y]. The classical van der Pol systems have f(y)=y. We first characterize when the origin of the generalized van der Pol systems is a center, and second we provide the global phase portraits in the Poincaré disc of the generalized van der Pol when f(y)=a₁y+a₂y² for all a₁,a₂∈R.

Original language	English
Article number	103213
Number of pages	19
Journal	Bulletin des Sciences Mathematiques
Volume	182
DOIs	https://doi.org/10.1016/j.bulsci.2022.103213
Publication status	Published - Feb 2023

Keywords

Blow ups
Center
Lyapunov constant
Poincaré compactification
van der Pol systems

Access to Document

10.1016/j.bulsci.2022.103213

https://ddd.uab.cat/record/274774

Cite this

@article{7b19a0e285f64794919d26ecbf3332c0,

title = "Global phase portraits of the generalized van der Pol systems",

abstract = "We consider the generalized van der Pol systems x˙=y,y˙=−x+(1−x2)f(y), where f∈R[y]. The classical van der Pol systems have f(y)=y. We first characterize when the origin of the generalized van der Pol systems is a center, and second we provide the global phase portraits in the Poincar{\'e} disc of the generalized van der Pol when f(y)=a1y+a2y2 for all a1,a2∈R.",

keywords = "Blow ups, Center, Lyapunov constant, Poincar{\'e} compactification, van der Pol systems",

author = "Jaume Llibre and Claudia Valls",

note = "Publisher Copyright: {\textcopyright} 2022 Elsevier Masson SAS",

year = "2023",

month = feb,

doi = "10.1016/j.bulsci.2022.103213",

language = "English",

volume = "182",

journal = "Bulletin des Sciences Mathematiques",

issn = "0007-4497",

publisher = "Elsevier Masson SAS",

}

TY - JOUR

T1 - Global phase portraits of the generalized van der Pol systems

AU - Llibre, Jaume

AU - Valls, Claudia

PY - 2023/2

Y1 - 2023/2

N2 - We consider the generalized van der Pol systems x˙=y,y˙=−x+(1−x2)f(y), where f∈R[y]. The classical van der Pol systems have f(y)=y. We first characterize when the origin of the generalized van der Pol systems is a center, and second we provide the global phase portraits in the Poincaré disc of the generalized van der Pol when f(y)=a1y+a2y2 for all a1,a2∈R.

AB - We consider the generalized van der Pol systems x˙=y,y˙=−x+(1−x2)f(y), where f∈R[y]. The classical van der Pol systems have f(y)=y. We first characterize when the origin of the generalized van der Pol systems is a center, and second we provide the global phase portraits in the Poincaré disc of the generalized van der Pol when f(y)=a1y+a2y2 for all a1,a2∈R.

KW - Blow ups

KW - Center

KW - Lyapunov constant

KW - Poincaré compactification

KW - van der Pol systems

UR - http://www.scopus.com/inward/record.url?scp=85142683846&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/904e2497-0e68-373e-add0-0eae9e07e6d4/

U2 - 10.1016/j.bulsci.2022.103213

DO - 10.1016/j.bulsci.2022.103213

M3 - Article

AN - SCOPUS:85142683846

SN - 0007-4497

VL - 182

JO - Bulletin des Sciences Mathematiques

JF - Bulletin des Sciences Mathematiques

M1 - 103213

ER -

Global phase portraits of the generalized van der Pol systems

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this