Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles by Maoan Han & Pei Yu

Author:Maoan Han & Pei Yu
Language: eng
Format: epub
Publisher: Springer London, London

Note that all the focus values v i ,i=0,1,2,3,4,5, are given in linear forms of the coefficients a i . Further, consider the reverse order perturbations, one by one: first on a 4 for v 4, then on a 3 for v 3, on a 2 for v 2, on a 1 for v 1, and finally on a 0 for v 0. Therefore, the perturbation procedure is straightforward. Since , one may perturb a 4=−9a 5 to a 4=−9a 5+ε 1 (0<ε 1≪1), and thus . Similarly, we may let a 3=14a 5−ε 2 (0<ε 2≪ε 1), and thus . This procedure can be followed until a 0 for v 0. The main results of this section are summarized in the following theorem.

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