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, , , or ) 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.
KeywordsNormal Form Real System Center Problem Center Variety Polynomial System
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