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In this paper we study resonances in two degrees of freedom, autonomous, hamil-tonian systems. Due to the presence of a symmetry condition on one of the degrees of freedom, we show that some of the resonances vanish as lower order resonances. After determining the size of the resonance domain, we investigate this order change of resonance in a rather… (More)

After reviewing a number of results from geometric singular perturbation theory, we give an example of a theorem for periodic solutions in a slow manifold. This is illustrated by examples involving the van der Pol-equation and a modified logistic equation. Regarding nonhyperbolic transitions we discuss a four-dimensional relaxation oscillation and also… (More)

The method of multiple timescales is widely used in engineering and mathematical physics. In this article we draw attention to the literature on the comparison of various perturbation methods. We indicate where we can obtain an advantage from the concept of timescales, and we present examples where the anticipation of timescales makes sense and cases where,… (More)

Persistence and bifurcations of Lyapunov manifolds can be studied by a combination of averaging-normalization and numerical bifurcation methods. This can be extended to infinite-dimensional cases when using suitable averaging theorems. The theory is applied to the case of a parametrically excited wave equation. We find fast dynamics in a finite, resonant… (More)

In this paper we study a Hamiltonian system with a spatially asymmetric potential. We are interested in the eeects on the dynamics when the potential becomes symmetric slowly in time. We focus on a highly simpliied non-trivial model problem (a metaphor) to be able to pursue explicit calculations as far as possible. Using the techniques of averaging and… (More)

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