Applied Mathematics Letters
Bendixson's Theorem [H. Ricardo, A Modem Introduction to Differential Equations, Houghton-Mifflin, New York, Boston, 2003] is useful in proving the non-existence of periodic orbits for planar systems
dx/dt = F(x, y), dy/dt = G (x, y)
in a simply connected domain D, where F, 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
dx/d tau = B(x, y)F(x, y), dy/d tau = B(x, y)G(x, y)
does, which makes the subcase (1) more tractable, when suitable non-zero B (x, y) which are 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). (c) 2006 Elsevier Ltd. All rights reserved.
Original Publication Citation
Cooke, C. H. (2006). A system equivalence related to Dulac's extension of Bendixson's negative theorem for planar dynamical systems. Applied Mathematics Letters, 19(11), 1291-1292. doi:10.1016/j.aml.2006.04.003
Cooke, Charlie H., "A System Equivalence Related to Dulac's Extension of Bendixson's Negative Theorem for Planar Dynamical Systems" (2006). Mathematics & Statistics Faculty Publications. 83.