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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
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