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Melnikov theory for finite-time vector fields. (English) Zbl 0962.37007

The authors develop a modified Melnikov theory, i.e. the investigation of system \(\frac{du}{dt}= f(u)+ \varepsilon h(t,u,\varepsilon)\), which for \(\varepsilon=0\) has a homoclinic orbit to a hyperbolic equilibrium and the perturbation is given on a finite but large (at least of order \(|\ln \varepsilon|\)) time interval. The vector field is extended for all time values (artificially, close to the unperturbed case). By the help of the Melnikov integral, results similar to the original theory are obtained. The approach is motivated from, and applied to the viscous dissipation of 2-dimensional vorticity conserving flows.

MSC:

37C29 Homoclinic and heteroclinic orbits for dynamical systems
76D09 Viscous-inviscid interaction
37C60 Nonautonomous smooth dynamical systems
34D10 Perturbations of ordinary differential equations
34C45 Invariant manifolds for ordinary differential equations
37G20 Hyperbolic singular points with homoclinic trajectories in dynamical systems
76B65 Rossby waves (MSC2010)
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