On the detuned 2:4 resonance

We consider families of Hamiltonian systems in two degrees of freedom with an equilibrium in 1:2 resonance. Under detuning, this "Fermi resonance" typically leads to normal modes losing their stability through period-doubling bifurcations. For cubic potentials this concerns the short axial orbits and in galactic dynamics the resulting stable periodic orbits are called "banana" orbits. Galactic potentials are symmetric with respect to the co-ordinate planes whence the potential -- and the normal form -- both have no cubic terms. This $\mathbb{Z}_2 \times \mathbb{Z}_2$-symmetry turns the 1:2 resonance into a higher order resonance and one therefore also speaks of the 2:4 resonance. In this paper we study the 2:4 resonance in its own right, not restricted to natural Hamiltonian systems where $H = T + V$ would consist of kinetic and (positional) potential energy. The short axial orbit then turns out to be dynamically stable everywhere except at a simultaneous bifurcation of banana and "anti-banana" orbits, while it is now the long axial orbit that loses and regains stability through two successive period-doubling bifurcations.Comment: 31 pages, 7 figures: On line first on Journal of Nonlinear Science (2020

Providing safe passage into a larger life: supporting clients’ transformational change through coaching

This paper reports coaches’ experiences of supporting clients undergoing transformational change. Qualitative research was carried out using semi-structured interviews with six coaches and these were analysed using interpretative phenomenological analysis (IPA). The findings suggest clearly identifiable ingredients for creating a transformational space and reveal how coaches can open up a larger life for clients while helping them overcome their fears. New insights were identified into the role of coaches’ spirituality and client readiness blended with the ability to take a long term view for retrospectively emerging transformations

Hybrid simulation-optimization methods: A taxonomy and discussion

The possibilities of combining simulation and optimization are vast and the appropriate design highly depends on the problem characteristics. Therefore, it is very important to have a good overview of the different approaches. The taxonomies and classifications proposed in the literature do not cover the complete range of methods and overlook some important criteria. We provide a taxonomy that aims at giving an overview of the full spectrum of current simulation-optimization approaches. Our study may guide researchers who want to use one of the existing methods, give insights into the cross-fertilization of the ideas applied in those methods and create a standard for a better communication in the scientific community. Future reviews can use the taxonomy here described to classify both general approaches and methods for specific application fields.The possibilities of combining simulation and optimization are vast and the appropriate design highly depends on the problem characteristics. Therefore, it is very important to have a good overview of the different approaches. The taxonomies and classifications proposed in the literature do not cover the complete range of methods and overlook some important criteria. We provide a taxonomy that aims at giving an overview of the full spectrum of current simulation-optimization approaches. Our study may guide researchers who want to use one of the existing methods, give insights into the cross-fertilization of the ideas applied in those methods and create a standard for a better communication in the scientific community. Future reviews can use the taxonomy here described to classify both general approaches and methods for specific application fields. (C) 2014 Elsevier B.V. All rights reserved