Evaluating  cyclicity of  cubic systems with algorithms of computational algebra

Viktor Levandovskyy; Gerhard  Pfister; Valery G. Romanovski

doi:10.3934/cpaa.2012.11.2023

Article Contents

2012, Volume 11, Issue 5: 2023-2035. Doi: 10.3934/cpaa.2012.11.2023

This issue Previous Article Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities Next Article Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities

Evaluating cyclicity of cubic systems with algorithms of computational algebra

1.
Lehrstuhl D für Mathematik, RWTH Aachen University, Templergraben 64, D-52062 Aachen
2.
Technische University of Kaiserslautern Fachbereich Mathematik Erwin-Schrödinger Str. 48,D-67653 Kaiserslautern
3.
CAMTP - Center for Applied Mathematics and Theoretical Physics,University of Maribor, Krekova 2 , SI-2000 Maribor

Received: May 31, 2011

Revised: November 30, 2011

Published: August 2012

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Abstract

We describe an algorithmic approach to studying limit cycle bifurcations in a neighborhood of an elementary center or focus of a polynomial system. Using it we obtain an upper bound for cyclicity of a family of cubic systems. Then using a theorem by Christopher [3] we study bifurcation of limit cycles from each component of the center variety. We obtain also the sharp bound for the cyclicity of a generic time-reversible cubic system.

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Mathematics Subject Classification: Primary: 34C07; Secondary: 34C23.

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