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A variational interpretation of Melnikov’s function and exponentionally small separatrix splitting. (English) Zbl 0810.34037

Salamon, Dietmar (ed.), Symplectic geometry. Based on lectures given at a workshop and conference held at the University of Warwick, GB in August 1990. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 192, 5-35 (1993).
The author studies the ordinary differential equation \(u''(t)= F(t/\varepsilon, u(t))\), \(\varepsilon> 0\), considered as a perturbation of the averaged equation \(u''= F_ 0(u)\). The nonlinearity \(F\) is analytic in \(u\) and 1-periodic in \(t\); \(F_ 0(u)= \int^ 1_ 0 F(t,u)dt\). Let \(\phi_ \varepsilon\), \(\phi_ 0\) denote the Poincaré maps of the associated first-order systems, respectively. Assume that \(\phi_ 0\) has a hyperbolic fixed point at the origin with a homoclinic orbit \(U(t)\) which is analytic in \(t\) (even for \(t\in \mathbb{C}\) with \(|\text{Im t}|\) small). The main result states that \(\phi_ \varepsilon\) has a hyperbolic fixed point \({\mathcal O}_ \varepsilon\) near the origin and a homoclinic point \(P_ \varepsilon\). Moreover, let \(\Gamma_ \varepsilon\) be the union of the two segments on the stable respectively unstable manifold of \(\phi_ \varepsilon\) connecting \({\mathcal O}_ \varepsilon\) and \(P_ \varepsilon\). Then the measure of the region bounded by \(\Gamma_ \varepsilon\) and \(\phi_ \varepsilon(\Gamma_ \varepsilon)\) is bounded above essentially by \(\exp(-1/\varepsilon)\).
For the proof the author interprets homoclinic orbits as critical points of a functional \(a_ \varepsilon\) and follows a variational formulation of Melnikov’s method. He applies this to the forced Duffing equation, where he can give an explicit formula for \(a_ \varepsilon\).
For the entire collection see [Zbl 0788.00062].

MSC:

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34C29 Averaging method for ordinary differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
37G99 Local and nonlocal bifurcation theory for dynamical systems
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