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The Center Problem

In: The Center and Cyclicity Problems

Author

Listed:
  • Valery Romanovski

    (University of Maribor, Center for Applied Mathematics & Theorectical Physics)

  • Douglas Shafer

    (University of North Carolina, Mathematics Dept.)

Abstract

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.

Suggested Citation

  • Valery Romanovski & Douglas Shafer, 2009. "The Center Problem," Springer Books, in: The Center and Cyclicity Problems, chapter 0, pages 1-86, Springer.
    • Handle: RePEc:spr:sprchp:978-0-8176-4727-8_3
    DOI: 10.1007/978-0-8176-4727-8_3
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