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A Note on “Uniqueness of Limit Cycles in a Liénard-Type System”

✍ Scribed by Robert E. Kooij; Sun Jianhua

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
255 KB
Volume
208
Category
Article
ISSN
0022-247X

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✦ Synopsis

proposed a theorem to guarantee the uniqueness of limit cycles for the generalized Lienard system Ž .

Ž . Ž . dxrdt s h y y F x , dyrdt s yg x . We will give a counterexample to their Ž . theorem. It will be shown that their theorem is valid only if F x is monotone on certain intervals. For this case we give a shorter proof and we also show that the Ž . limit cycle is hyperbolic. If the condition on the monotonicity of F x is violated then we need an additional condition to guarantee the uniqueness of the limit cycle. Examples are provided to illustrate our results.

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We consider a class of planar differential equations which include the Lie´nard differential equations. By applying the Bendixson-Dulac Criterion for '-connected sets we reduce the study of the number of limit cycles for such equations to the condition that a certain function of just one variable do