Probability of local bifurcation type from a fixed point: A random matrix perspective

Results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems (real, random matrices), time-delay dynamical systems (companion matrices), and a set of mappings known for their properties as universal approximators (neural networks). The eigenvalue spectra is considered both numerically and analytically using previous work of Edelman et. al. Based upon the numerical evidence, various conjectures are presented. The conclusion is that in many circumstances, most bifurcations from fixed points of large dynamical systems will be due to complex eigenvalues. Nevertheless, surprising situations are presented for which the aforementioned conclusion is not general, e.g. real random matrices with Gaussian elements with a large positive mean and finite variance.Comment: 21 pages, 19 figure

Reaching micro-arcsecond astrometry with long baseline optical interferometry; application to the GRAVITY instrument

A basic principle of long baseline interferometry is that an optical path difference (OPD) directly translates into an astrometric measurement. In the simplest case, the OPD is equal to the scalar product between the vector linking the two telescopes and the normalized vector pointing toward the star. However, a too simple interpretation of this scalar product leads to seemingly conflicting results, called here "the baseline paradox". For micro-arcsecond accuracy astrometry, we have to model in full the metrology measurement. It involves a complex system subject to many optical effects: from pure baseline errors to static, quasi-static and high order optical aberrations. The goal of this paper is to present the strategy used by the "General Relativity Analysis via VLT InTerferometrY" instrument (GRAVITY) to minimize the biases introduced by these defects. It is possible to give an analytical formula on how the baselines and tip-tilt errors affect the astrometric measurement. This formula depends on the limit-points of three type of baselines: the wide-angle baseline, the narrow-angle baseline, and the imaging baseline. We also, numerically, include non-common path higher-order aberrations, whose amplitude were measured during technical time at the Very Large Telescope Interferometer. We end by simulating the influence of high-order common-path aberrations due to atmospheric residuals calculated from a Monte-Carlo simulation tool for Adaptive optics systems. The result of this work is an error budget of the biases caused by the multiple optical imperfections, including optical dispersion. We show that the beam stabilization through both focal and pupil tracking is crucial to the GRAVITY system. Assuming the instrument pupil is stabilized at a 4 cm level on M1, and a field tracking below 0.2$\lambda/D$, we show that GRAVITY will be able to reach its objective of 10$\mu$as accuracy.Comment: 14 pages. Accepted by A&

Improving Stability Prediction in Peripheral Milling of Al7075T6

Chatter is an old enemy to machinists but, even today, is far from being defeated. Current requirements around aerospace components call for stronger and thinner workpieces which are more prone to vibrations. This study presents the stability analysis for a single degree of freedom down-milling operation in a thin-walled workpiece. The stability charts were computed by means of the enhanced multistage homotopy perturbation (EMHP) method, which includes the helix angle but also, most importantly, the runout and cutting speed effects. Our experimental validation shows the importance of this kind of analysis through a comparison with a common analysis without them, especially when machining aluminum alloys. The proposed analysis demands more computation time, since it includes the calculation of cutting forces for each combination of axial depth of cut and spindle speed. This EMHP algorithm is compared with the semi-discretization, Chebyshev collocation, and full-discretization methods in terms of convergence and computation efficiency, and ultimately proves to be the most efficient method among the ones studied.The authors wish to acknowledge the financial support received from HAZITEK program, from the Department of Economic Development and Infrastructures of the Basque Government and from FEDER funds. Additional support was provided by the Tecnologico de Monterrey, through the Research Group in Nanomaterials and Devices Design

On Norm-Based Estimations for Domains of Attraction in Nonlinear Time-Delay Systems

For nonlinear time-delay systems, domains of attraction are rarely studied despite their importance for technological applications. The present paper provides methodological hints for the determination of an upper bound on the radius of attraction by numerical means. Thereby, the respective Banach space for initial functions has to be selected and primary initial functions have to be chosen. The latter are used in time-forward simulations to determine a first upper bound on the radius of attraction. Thereafter, this upper bound is refined by secondary initial functions, which result a posteriori from the preceding simulations. Additionally, a bifurcation analysis should be undertaken. This analysis results in a possible improvement of the previous estimation. An example of a time-delayed swing equation demonstrates the various aspects.Comment: 33 pages, 8 figures, "This is a pre-print of an article published in 'Nonlinear Dynamics'. The final authenticated version is available online at https://doi.org/10.1007/s11071-020-05620-8