Abstract
We consider the class of polynomial differential equations x = λx - y + Pn(x, y), y = x + λy + Qn (x, y), where Pn and Qn are homogeneous polynomials of degree n. These systems have a focus at the origin if λ ≠ 0, and have either a center or a focus if λ = 0. Inside this class we identify a new subclass of Darbouxian integrable systems having either a focus or a center at the origin. Additionally, under generic conditions such Darbouxian integrable systems can have at most one limit cycle, and when it exists is algebraic. For the case n = 2 and 3, we present new classes of Darbouxian integrable systems having a focus. © 2003 Elsevier Inc. All rights reserved.
Original language | English |
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Pages (from-to) | 147-161 |
Journal | Journal of Differential Equations |
Volume | 197 |
Issue number | 1 |
DOIs | |
Publication status | Published - 10 Feb 2004 |
Keywords
- Algebraic limit cycle
- Center
- Focus
- Integrability