#### Document Type

Article

#### Publication Date

2006

#### DOI

10.1016/j.aml.2006.04.003

#### Publication Title

Applied Mathematics Letters

#### Volume

19

#### Issue

11

#### Pages

1291-1292

#### Abstract

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 C^{1}(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

#### Repository Citation

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.

https://digitalcommons.odu.edu/mathstat_fac_pubs/83

## Comments

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Copyright © 2006 Elsevier Ltd. All rights reserved.