Melnikov theory for weakly coupled nonlinear RLC circuits

We apply dynamical system methods and Melnikov theory to study small amplitude perturbation of coupled nonlinear RLC systems. In particular we show persistence of such orbits connecting singularities in finite time provided a Melnikov like condition holds

Melnikov theory for nonlinear implicit ODEs

Abstract We apply dynamical system methods and Melnikov theory to study small amplitude perturbation of some implicit differential equations. In particular we show persistence of such orbits connecting singularities in finite time provided a Melnikov like condition holds

Homoclinic orbits of slowly periodically forced and weakly damped beams resting on weakly elastic bearings

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Chaos arising near a topologically transversal homoclinic set

A diffeomorphism on a $C^1$-smooth manifold is studied possessing a hyperbolic fixed point. If the stable and unstable manifolds of the hyperbolic fixed point have a nontrivial local topological crossing then a chaotic behaviour of the diffeomorphism is shown. A perturbed problem is also studied by showing the relationship between a corresponding Melnikov function and the nontriviality of a local topological crossing of invariant manifolds for the perturbed diffeomorphism