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- Ferdinand Verhulst
- Encyclopedia of Complexity and Systems Science
- 2009

The paradox of destabilization of a conservative or non-conservative system by small dissipation, or Ziegler's paradox (1952), has stimulated an ever growing interest in the sensitivity of reversible and Hamiltonian systems with respect to dissipative perturbations. Since the last decade it has been widely accepted that dissipation-induced instabilities are… (More)

- Ferdinand Verhulst, J. M. Tuwankotta
- SIAM Journal of Applied Mathematics
- 2001

In this paper we study resonances in two degrees of freedom, autonomous, Hamiltonian 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 giving a sharp estimate of the resonance domain, we investigate this order change of resonance in a rather… (More)

Stable normal mode vibrations in engineering can be undesirable and one of the possibilities for quenching these vibrations is by embedding the oscillator in an autoparametric system by coupling to a damped oscillator. We have the possibility of destabilising the undesirable vibrations by a suitable tuning and choice of coupling parameters. In the case of… (More)

In this paper we study two degree of freedom Hamiltonian systems and applications to nonlinear wave equations. Near the origin, we assume that near the linearized system has purely imaginary eigenvalues: i! 1 and i! 2 , with 0 < ! 2 =! 1 1 or ! 2 =! 1 1, which is interpreted as a perturbation of a problem with double zero eigenvalues. Using the averaging… (More)

- Cor van Dijkum, Dorien J. DeTombe, +7 authors Dorien J. DeTombe
- 1999

©1999 No part of this book may be reproduced in any form by print, photoprint, microfilm or any other means without the prior written permission from the publisher. A publication in coöperation with the Dutch researchgroups SIMULATION and COMPLEX SOCIETAL PROBLEMS of the NOSMO (Dutch Research group on Methodology) Validation of simulation models in the… (More)

Invariant manifolds like tori, spheres and cylinders play an important part in dynamical systems. In engineering, tori correspond with the important phenomenon of multi-frequency oscillations. Normal hyperbolicity guarantees the robustness of these manifolds but in many applications weaker forms of hyperbolicity present more realistic cases and interesting… (More)

- Ferdinand Verhulst
- I. J. Bifurcation and Chaos
- 2007

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)