Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom

This paper concerns Hamiltonian and non-Hamiltonian perturbations of integrable two degree of freedom Hamiltonian systems which contain homoclinic and periodic orbits. Our main example concerns perturbations of the uncoupled system consisting of the simple pendulum and the harmonic oscillator. We show that small coupling perturbations with, possibly, the addition of positive and negative damping breaks the integrability by introducing horseshoes into the dynamics.Research partially supported by ARO Contract DAAG-29-79-C-0086 and by NSF Grants ENG 78-02891 and MCS-78-06718     

The mass spectra of carboxylic acids—I: Fragmentation mechanisms in maleic and fumaric acids and related compounds

The mass spectra of maleic acid, maleic acid-2,3-d, fumaric acid and fumaric acid-2,3-d have been examined and fragmentation mechanisms are proposed for these compounds. The molecular ion of the cis-acid fragments via H atom transfer from one carboxyl group to the other followed by loss of CO₂. The trans acid does not fragment significantly by this route and the former effect may be characteristic of molecules containing two carboxyl groups cis-oriented to each other. This hypothesis was successfully tested by examining the mass spectra of citraconic, itaconic and phthalic acids. Itaconic and mesaconic acids show some of the fragmentation characteristics of fumaric acid.     

‘Strange’ phenomena in dynamical systems and their physical implications

Some recent developments in dynamical systems theory are outlined and their physical implications are discussed. In particular we introduce the concept of ‘strange attractors’: motions which arise as solutions of deterministic dynamical systems but which have extremely complicated and apparently random structures. We suggest that the behaviour of such motions has serious consequences for dynamical modelling exercises.     

Bifurcations to divergence and flutter in flow-induced oscillations: A finite dimensional analysis

The behaviours of a pipe conveying fluid and a fluid loaded panel are studied from the viewpoint of differentiable dynamics. Non-linear terms are included and it is shown how the partial differential equation of motion can be recast, by Galerkin's method and modal truncation, in the form of an ordinary differential equation in Euclidean n-space. This evolution equation is then analysed qualitatively, attention being paid to bifurcations which occur as the control parameters of axial force and flow velocity are varied. Bifurcations of fixed points occur when at least one of the eigenvalues of the linearized evolution equation crosses the imaginary axis in the complex plane. In this situation, centre manifold theory can be used to extract a low dimensional subsystem which completely captures the local bifurcational behaviour. Such essential models enable the onset of divergence and flutter to be analysed relatively simply and the inclusion of non-linear terms permits the global study of post-bifurcational behaviour. The general approach is illustrated by analysis of two mode models of a pipe and of a panel and some important omissions in previous treatments of linear and undamped systems are discussed.     

The experimental characterization of wave propagation systems: II. Continuous systems and the effects of dispersion

This paper forms the second part of a study of the wave propagation approach and its application to the experimental characterization of structural/acoustic systems. It is shown that the measurement of time delays and attenuation factors by correlation techniques offers a useful complement to the measurement of frequency response in system characterization. Transient excitation is used and although the rapid frequency sweep is suitable for measurements in continuous systems, it is shown that the dispersion encountered in flexural wave propagation increases the difficulty in the interpretation of correlation data. To overcome this problem, another transient excitation signal is developed which has essentially single frequency content but which yields clear correlation functions, permitting the accurate measurement of group velocities at a single frequency.