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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)

MSC:
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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