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The Center Problem

  • Valery Romanovski
  • Douglas Shafer
Chapter

Consider a real planar system of differential equations u˙= f(u), deӿned and analytic on a neighborhood of 0, for which f(0)= 0and the eigenvalues of the linear part of fat 0are α ± iβ with β =0. If the system is actually linear, then a straightforward geometric analysis (see, for example, [44], [95], or [110]) shows that when α = 0

the trajectory of every point spirals towards or away from 0(see Deӿnition 3.1.1: 0is a focus), but when α = 0, the trajectory of every point except 0is a cycle, that is, lies in an oval (see Deӿnition 3.1.1: 0is a center). When the system is nonlinear, then in the ӿrst case (α = 0) trajectories in a sufӿciently small neighborhood of the

origin follow the behavior of the linear system determined by the linear part of fat

0: they spiral towards or away from the origin in accordance with the trajectories of the linear system.

Keywords

Normal Form Real System Center Problem Center Variety Polynomial System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Center for Applied Mathematics & Theorectical PhysicsUniversity of MariborSlovenia
  2. 2.Mathematics Dept.University of North CarolinaCharlotteUSA

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