Abstract
© 2015 Elsevier Inc. We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S admits at most S-1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S-1 algebraic limit cycles given by circles, and this number is reached.
Original language | English |
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Pages (from-to) | 5726-5760 |
Journal | Journal of Differential Equations |
Volume | 260 |
Issue number | 7 |
DOIs | |
Publication status | Published - 5 Apr 2016 |
Keywords
- 16th Hilbert's problem
- Algebraic limit circles
- Darboux integrability
- Invariant algebraic circles
- Planar polynomial differential system
- Polynomial vector fields