A proof of Perko's conjectures for the Bogdanov-Takens system

A. Gasull; H. Giacomini; S. Pérez-González; J. Torregrosa

doi:https://doi.org/10.1016/j.jde.2013.07.006

A proof of Perko's conjectures for the Bogdanov-Takens system

A. Gasull, H. Giacomini, S. Pérez-González, J. Torregrosa

Department of Mathematics

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

7 Citations (Scopus)

Abstract

The Bogdanov-Takens system has at most one limit cycle and, in the parameter space, it exists between a Hopf and a saddle-loop bifurcation curves. The aim of this paper is to prove the Perko's conjectures about some analytic properties of the saddle-loop bifurcation curve. Moreover, we provide sharp piecewise algebraic upper and lower bounds for this curve. © 2013 Elsevier Inc.

Original language	English
Pages (from-to)	2655-2671
Journal	Journal of Differential Equations
Volume	255
Issue number	9
DOIs	https://doi.org/10.1016/j.jde.2013.07.006
Publication status	Published - 1 Nov 2013

Keywords

Bifurcation of limit cycles
Global description of bifurcation curve
Homoclinic connection
Location of limit cycles
Primary
Secondary

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abstract = "The Bogdanov-Takens system has at most one limit cycle and, in the parameter space, it exists between a Hopf and a saddle-loop bifurcation curves. The aim of this paper is to prove the Perko's conjectures about some analytic properties of the saddle-loop bifurcation curve. Moreover, we provide sharp piecewise algebraic upper and lower bounds for this curve. {\textcopyright} 2013 Elsevier Inc.",

keywords = "Bifurcation of limit cycles, Global description of bifurcation curve, Homoclinic connection, Location of limit cycles, Primary, Secondary",

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AU - Gasull, A.

AU - Giacomini, H.

AU - Pérez-González, S.

AU - Torregrosa, J.

PY - 2013/11/1

Y1 - 2013/11/1

N2 - The Bogdanov-Takens system has at most one limit cycle and, in the parameter space, it exists between a Hopf and a saddle-loop bifurcation curves. The aim of this paper is to prove the Perko's conjectures about some analytic properties of the saddle-loop bifurcation curve. Moreover, we provide sharp piecewise algebraic upper and lower bounds for this curve. © 2013 Elsevier Inc.

AB - The Bogdanov-Takens system has at most one limit cycle and, in the parameter space, it exists between a Hopf and a saddle-loop bifurcation curves. The aim of this paper is to prove the Perko's conjectures about some analytic properties of the saddle-loop bifurcation curve. Moreover, we provide sharp piecewise algebraic upper and lower bounds for this curve. © 2013 Elsevier Inc.

KW - Bifurcation of limit cycles

KW - Global description of bifurcation curve

KW - Homoclinic connection

KW - Location of limit cycles

KW - Primary

KW - Secondary

U2 - https://doi.org/10.1016/j.jde.2013.07.006

DO - https://doi.org/10.1016/j.jde.2013.07.006

M3 - Article

SN - 0022-0396

VL - 255

SP - 2655

EP - 2671

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 9

ER -

A proof of Perko's conjectures for the Bogdanov-Takens system

Abstract

Keywords

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