Ferdinand Verhulst

Paradoxes of dissipation-induced destabilization or who opened Whitney's umbrella?

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 closely related to singularities arising on the stability boundary. What is less known is that the first complete explanation of Ziegler’s paradox by means of the Whitney umbrella singularity dates back to 1956. We revisit this undeservedly forgotten pioneering result by Oene Bottema that outstripped later findings for about half a century. We discuss subsequent developments of the perturbation analysis of dissipation-induced instabilities and applications over this period, involving structural stability of matrices, Krein collision, Hamilton-Hopf bifurcation, and related bifurcations. c

Discrete Symmetric Dynamical Systems at the Main Resonances with Applications to Axi-Symmetric Galaxies

A study of two-degrees-of-freedom systems with a potential which is discrete-symmetric (even in one of the position variables) is carried out for the resonance cases 1:2, 1:1, 2:1 and 1:3. To produce both qualitative and quantitative results, we obtain in each resonance case normal forms by higher order averaging procedures. This method is related to Birkhoff normalization and provides us with rigorous asymptotic estimates for the approximate solutions. The normal forms have been used to obtain a classification of possible local and global bifurcations for these dynamical systems. One of the applications here is to describe the two-parameter family of bifurcations obtained by detuning a one-parameter family studied by Braun. In all the resonances discussed an approximate integral of the motion other than the total energy exists, but in the 2:1 and 1:3 resonance cases this degenerates into the partial energy of the z motion. In conclusion some remarks are made on the relation between two-degrees-of-freedom systems and solutions of the collisionless Boltzmann equation. Moreover we are able to make some observations on the Henon-Heiles problem and certain classical examples of potentials.

Near-integrability of periodic FPU-chains

The FPU-chain with periodic boundary conditions is studied by Birkhoff–Gustavson normalisation. In the cases of up to 6 particles and for β-chains with an odd number of particles the normal forms are integrable, which permits us to apply KAM-theory. This leads to the presence of many invariant tori on which the motion is quasi-periodic. Thus we explain the recurrence phenomena and the small size of chaos observed in experiments. Furthermore, we find a certain clustering of modes.

Periodic Solutions and Slow Manifolds

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 canard-like solutions emerging from the modified logistic equation with sign-alternating growth rates.

Symmetry and resonance in Hamiltonian systems

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 general potential problem with discrete symmetry and consider as an example the Henon--Heiles family of Hamiltonians. We also study a classical example of a mechanical system with symmetry, the elastic pendulum, which leads to a natural hierarchy of resonances with the 4:1-resonance as the most prominent after the 2:1-resonance and which explains why the 3:1-resonance is neglected.

Chaos in the 1:2:3 Hamiltonian normal form

The normal form of the Hamiltonian 1:2:3 resonance to degree 3 contains seven families of periodic solutions of which one can be complex unstable. Associated with this complex unstable solution is an invariant manifold N on which the dynamics can be characterised completely; one of the ingredients of N is a set of homoclinic orbits. In the normal form to degree 4 the set of homoclinic orbits breaks up, for certain parameter values, into one homoclinic orbit. This enables us to apply Silnikov-Devaney theory to prove, at this stage numerically, the existence of a horseshoe map in the system with the implication of non-integrability and chaos in the normal form.

Parametric excitation in non-linear dynamics

Abstract Consider a one-mass system with two degrees of freedom, non-linearly coupled, with parametric excitation in one direction. Assuming the internal resonance 1:2 and parametric resonance 1:2 we derive conditions for stability of the trivial solution by using both the harmonic balance method and the normal form method of averaging. If the trivial solution becomes unstable, a stable periodic solution may emerge, there are also cases where the trivial solution is stable and co-exists with a stable periodic solution; if both the trivial solution and the periodic solution(s) are unstable, we find an attracting torus with large amplitudes by a Neimark–Sacker bifurcation. The results of the harmonic balance method and averaging are compared, as well as the results on the Neimark–Sacker bifurcation obtained by the numerical software package CONTENT and by averaging. In all cases we have good agreement.

Suppressing Flow-Induced Vibrations by Parametric Excitation

The possibility of suppressing self-excited vibrations of mechanicalsystems using parametric excitation is discussed. We consider a two-masssystem of which the main mass is excited by a flow-induced, self excitedforce. A single mass which acts as a dynamic absorber is attached to themain mass and, by varying the stiffness between the main mass and theabsorber mass, represents a parametric excitation. It turns out that forcertain parameter ranges full vibration cancellation is possible. Usingthe averaging method the fully non-linear system is investigatedproducing as non-trivial solutions stable periodic solutions and tori.In the case of a small absorber mass we have to carry out a second-ordercalculation.

The method of averaging

In this chapter we shall consider again equations containing a small parameter e. The approximation method leads generally to asymptotic series as opposed to the convergent series studied in the preceding chapter; see section 9.2 for the basic concepts and more discussion in Sanders and Verhulst (1985), chapter 2. This asymptotic character of the approximations is more natural in many problems; also the method turns out to be very powerful, it is not restricted to periodic solutions.

Asymptotic Integrability and Periodic Solutions of a Hamiltonian System in

A Hamiltonian system in