A system equivalence related to Dulac’s extension of Bendixson’s negative theorem for planar dynamical systems Original Research Article

Bendixson’s Theorem [H. Ricardo, A Modern Introduction to Differential Equations, Houghton-Mifflin, New York, Boston, 2003] is useful in proving the non-existence of periodic orbits for planar systems equation(1) View the MathML sourcedxdt=F(x,y),dydt=G(x,y) Turn MathJax on in a simply connected domain DD, where F,GF,G are continuously differentiable. From the work of Dulac [M. Kot, Elements of Mathematical Ecology, 2nd printing, University Press, Cambridge, 2003] one suspects that system (1) has periodic solutions if and only if the more general system equation(2) View the MathML sourcedxdτ=B(x,y)F(x,y),dydτ=B(x,y)G(x,y) Turn MathJax on does, which makes the subcase (1) more tractable, when suitable non-zero B(x,y)B(x,y) which are C1(D)C1(D) can be found. Thus, Bendixson’s Theorem can be applied to system (2), where otherwise it is unfruitful in establishing the non-existence of periodic solutions for system (1). The object of this note is to give a simple proof justifying this Dulac-related postulate of the equivalence of systems (1) and (2).