On a Dulac function for the Kukles system

Some Applications of the Extended Bendixson-Dulac Theorem

“…and fix a positive integer number n. There is a constructive procedure to obtain an (n + 1)-th order linear differential equation y (n+1) (x) + r n,n (x) y (n) (x) + · · · + r n,1 (x) y ′ (x) + r n,0 (x) y(x) = 0, (9) such that if y(x) = v n (x) is any of its solutions, we can define a function V n (x, y) := v n,0 (x) + v n,1 (x)y + v n,2 (x)y 2 + · · · + v n,n (x)y n ,…”

“…. By imposing that this last expression be identically zero we get the linear ordinary differential equation (9) given in the statement of the lemma. Hence for these values of the functions v i , i = 0, 1, 2 the expression of M [2] is the function of one variable M [2]…”

“…We can not end this introduction without talking about the contributions on the use of Dulac functions of our friend and colleague Leonid Cherkas, sadly recently deceased. His important work in this subject started many years ago and arrives until the actuality, continued by his collaborators, see for instance [4,5,6,7,8,9,10] and the reference therein. In fact, one of the main motivations for the fist author to work in this direction were the pleasant conversations with him walking around the beautiful gardens of the Beijing University in the summer of 1990.…”

On Muldowney’s Criteria for Polynomial Vector Fields with Constraints

“…A common class of Dulac functions uses B(x, y) = e (U (x,y)) , see e.g. [3]. By the chain rule the exponential function can be factored out yielding e U ∂U ∂x F + ∂U ∂y G + ∂F ∂x + ∂G ∂y .…”

“…This criterion is parameterized by a Dulac function, and various techniques have been proposed to construct Dulac functions, which range form algebraic constructions for special systems to techniques involving the solution of certain partial differential equations [2][3][4][5][6].…”