Bifurcation diagram and stability for a one-parameter family of planar vector fields

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:413

Article

Garcia-Saldana, J. D.¹  Gasull, A.¹  Giacomini, H.² 

[1] Univ Autonoma Barcelona, Dept Matemat, E-08193 Barcelona, Spain

[2] Univ Tours, CNRS, Fac Sci & Tech, Lab Math & Phys Theor,UMR 7350, F-37200 Tours, France

关键词: Planar polynomial system;    Uniqueness and hyperbolicity of the limit cycle;    Polycycle;    Bifurcation;    Phase portrait on the Poincare disc;    Dulac function;    Stability;    Nilpotent point;    Basin of attraction;   

DOI  :  10.1016/j.jmaa.2013.11.047

来源: Elsevier

【 摘 要 】

We consider the one-parameter family of planar quintic systems, (x) over dot = y(3) - x(3), (y) over dot = -x + my(5), introduced by A. Bacciotti in 1985. It is known that it has at most one limit cycle and that it can exist only when the parameter m is in (0.36, 0.6). In this paper, using the Bendixson-Dulac theorem, we give a new unified proof of all the previous results. We shrink this interval to (0.547, 0.6) and we prove the hyperbolicity of the limit cycle. Furthermore, we consider the question of the existence of polycycles. The main interest and difficulty for studying this family is that it is not a semi-complete family of rotated vector fields. When the system has a limit cycle, we also determine explicit lower bounds of the basin of attraction of the origin. Finally, we answer an open question about the change of stability of the origin for an extension of the above systems. (C) 2013 Elsevier Inc. All rights reserved.

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