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A System Equivalence Related To Dulac's Extension Of Bendixson's Negative Theorem For Planar Dynamical Systems, Charlie H. Cooke

Mathematics & Statistics Faculty Publications

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 …