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Subharmonic solutions in singular systems. (English) Zbl 0887.34029
The authors consider the problem of bifurcation of periodic solutions in singular systems of differential equations \[ \varepsilon\dot{u}=f(u)+\varepsilon g(t,u,\varepsilon)\quad u\in\mathbb{R}^n, \] where \(g(t+2,u,\varepsilon)=g(t,u,\varepsilon)\) and \(\dot{u}=f(u)\) has an orbit \(\gamma(t)\) homoclinic to a hyperbolic equilibrium point \(p\). By using a functional analytic approach and the Lyapunov-Schmidt method they obtain a bifurcation function which tends, as \(\varepsilon\to0+\), to the Melnikov function. They show that if a certain Melnikov condition is satisfied then the system has a unique periodic solution of period \(2m\), for any \(m\geq1\), \(m\in\mathbb{N}\), and \(\varepsilon\) sufficiently small.
Reviewer: A.Reinfelds (Riga)

34C23 Bifurcation theory for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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