A system equivalence related to Dulac’s extension of Bendixson’s negative theorem for planar dynamical systems
| Autor: | Charlie H. Cooke |
|---|---|
| Jazyk: | angličtina |
| Předmět: |
Pure mathematics
Dynamical systems theory Differential equation Applied Mathematics Mathematical analysis Domain (mathematical analysis) Parameterized system equivalence Bendixson–Dulac Theorem Simple (abstract algebra) Bendixson–Dulac theorem Periodic solutions of planar systems Simply connected space Dynamical system (definition) System equivalence Mathematics |
| Zdroj: | Applied Mathematics Letters. (11):1291-1292 |
| ISSN: | 0893-9659 |
| DOI: | 10.1016/j.aml.2006.04.003 |
| Popis: | 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 (1) d x d t = F ( x , y ) , d y d t = 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 (2) d x d τ = B ( x , y ) F ( x , y ) , d y d τ = 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) , (2) . |
| Databáze: | OpenAIRE |
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