Evaluating cyclicity of cubic systems with algorithms of computational algebra

Pages: 2023 - 2035, Volume 11, Issue 5, September 2012

doi:10.3934/cpaa.2012.11.2023       Abstract        References        Full Text (423.7K)       Related Articles

Viktor Levandovskyy - Lehrstuhl D für Mathematik, RWTH Aachen University, Templergraben 64, D-52062 Aachen, Germany (email)
Gerhard Pfister - Technische University of Kaiserslautern Fachbereich Mathematik Erwin-Schrödinger Str. 48,D-67653 Kaiserslautern, Germany (email)
G. Romanovskiy - CAMTP - Center for Applied Mathematics and Theoretical Physics,University of Maribor, Krekova 2 , SI-2000 Maribor, Slovenia (email)

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.

Keywords:  Limit cycles, bifurcation, cyclicity, polynomial systems of ODEs.
Mathematics Subject Classification:  Primary: 34C07; Secondary: 34C23.

Received: June 2011;      Revised: December 2011;      Published: March 2012.

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