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Some remarks on the Melnikov function. (English) Zbl 1002.34034
The authors study the system $x'=f(x)+\varepsilon h(t+\alpha,x,\varepsilon),$ where $$\varepsilon$$ is a small parameter. It is assumed that if $$\varepsilon=0$$, then the system has a nondegenerate homoclinic solution $$\phi(t)$$. The Melnikov function $$M(\alpha)$$ is studied in the case where $$\phi(t)=\Phi(e^t)$$ for a rational function $$\Phi$$. The authors also study the second-order Melnikov function $$M_2(\alpha)$$, i.e., the coefficient at $$\varepsilon^2/2$$ in the expansion of the bifurcation function.

##### MSC:
 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations
##### Keywords:
Melnikov function; residues; Fourier coefficients; bifurcation
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