Global structure of ordinary differential equations in the plane

Quadratic Planar Systems with Two Parallel Invariant Straight Lines

“…Neumann' s Theorem was obtained under the additional assumption that the flow has no limit separatrices by Markus [10] …”

“…In the same way we have the global phase portraits for the cases k = 1, 2, 3, 4, 5, 6,7,8,10,11,12,13,14,15,16,17. (5) is topologically equivalent to one of the 27 configurations of Figure 9 labeled A n with n = 1, .…”

“…See for example the cases D n where n ∈ {1, 2, 3, 4, 5, 6, 7, 9}. (19) is topologically equivalent to one of the 39 configurations of Figures 9 and 10 labeled C n with n = 1, 2, 3, 4, 5, 6,7,8,9,10,11,12,13,19,20,21,22,23,24,25,26,27,28,29, 30, 31, 32, 34, 36, 37 and D n with n = 1, . .…”

“…The global phase portraits of the vector field (20) is topologically equivalent to one of the 19 configurations ofFigure 10labeled C n with n = 7,8,9,10,11,12,13, 25, 26, 27, 28, 29, 30, D n = 3, 6, 7, 8, 9 and E 1 .…”

Solvability of the cohomological equation for regular vector fields on the plane

“…A leaf s is called a separatrix when the boundary of every neighbourhood of π ξ (s) contains more than two points. The set of all separatrices is the closure of S F ξ [12]. In the present article, we will rather use the term separatrix to indicate just the elements of S F ξ since their limit points play no role in our study.…”