Dynamic Stability Analysis of a Hydrofoiling Sailing Boat using CFD

This paper describes the study of the dynamic stability of a hydrofoiling sailing boat called the “Goodall Design Foiling Viper”. The goal of Goodall Design is to make hydrofoiling accessible to a wider public, whereas it was previously reserved for professional sailors at the highest level of the sport. To allow for safe operation, stability is an essential characteristic of the boat. The aim of this work is to find a strategy to perform a dynamic stability analysis using computational fluid dynamics (CFD), which can be used in a preliminary design stage. This paper starts by establishing a theoretical framework to perform the dynamic stability analysis. A stability analysis has to be performed around an equilibrium state which depends on operating parameters such as speed, centre of gravity, etc. . A fluid-structure interaction strategy is applied to determine these equilibrium states. The last part discusses the stability characteristics of the Viper. The framework managed to assess the dynamic stability of the Viper and found 5 longitudinal eigenmodes: two complex conjugated pairs of eigenvalues and one real eigenvalue. It can be concluded that the boat was both statically and dynamically stable.

Stability and dynamic transition of vegetation model for flat arid terrains

<p style='text-indent:20px;'>In this paper, we aim to investigate the dynamic transition of the Klausmeier-Gray-Scott (KGS) model in a rectangular domain or a square domain. Our research tool is the dynamic transition theory for the dissipative system. Firstly, we verify the principle of exchange of stability (PES) by analyzing the spectrum of the linear part of the model. Secondly, by utilizing the method of center manifold reduction, we show that the model undergoes a continuous transition or a jump transition. For the model in a rectangular domain, we discuss the transitions of the model from a real simple eigenvalue and a pair of simple complex eigenvalues. our results imply that the model bifurcates to exactly two new steady state solutions or a periodic solution, whose stability is determined by a non-dimensional coefficient. For the model in a square domain, we only focus on the transition from a real eigenvalue with algebraic multiplicity 2. The result shows that the model may bifurcate to an <inline-formula><tex-math id="M1">\begin{document}$ S^{1} $\end{document}</tex-math></inline-formula> attractor with 8 non-degenerate singular points. In addition, a saddle-node bifurcation is also possible. At the end of the article, some numerical results are performed to illustrate our conclusions.</p>

Eigenvalues of the non-backtracking operator detached from the bulk

We describe the non-backtracking spectrum of a stochastic block model with connection probabilities [Formula: see text]. In this regime we answer a question posed in [L. Dall’Amico, R. Couillet and N. Tremblay, Revisiting the Bethe–Hessian: Improved community detection in sparse heterogeneous graphs, in Advances in Neural Information Processing Systems (2019), pp. 4039–4049] regarding the existence of a real eigenvalue “inside” the bulk, close to the location [Formula: see text]. We also introduce a variant of the Bauer–Fike theorem well suited for perturbations of quadratic eigenvalue problems, which could be of independent interest.

Existence of Chaos in the Chen System with Linear Time-Delay Feedback

It is mathematically challenging to analytically show that complex dynamical phenomena observed in simulations and experiments are truly chaotic. The Shil’nikov lemma provides a useful theoretical tool to prove the existence of chaos in three-dimensional smooth autonomous systems. It requires, however, the proof of existence of a homoclinic or heteroclinic orbit, which remains a very difficult technical problem if contigent on data. In this paper, for the Chen system with linear time-delay feedback, we demonstrate a homoclinic orbit by using a modified undetermined coefficient method and we propose a spiral involute projection method. In such a way, we identify experimentally the asymmetrical homoclinic orbit in order to apply the Shil’nikov-type lemma and to show that chaos is indeed generated in the Chen circuit with linear time-delay feedback. We also identify the presence of a single-scroll attractor in the Chen system with linear time-delay feedback in our experiments. We confirm that the Chen single-scroll attractor is hyperchaotic by numerically estimating the finite-time local Lyapunov exponent spectrum. By means of a linear scaling in the coordinates and the time, such a method can also be applied to the generalized Lorenz-like systems. The contribution of this work lies in: first, we treat the trajectories corresponding to the real eigenvalue and the image eigenvalues in different ways, which is compatible with the characteristics of the trajectory geometry; second, we propose a spiral involute projection method to exhibit the trajectory corresponding to the image eigenvalues; third, we verify the homoclinic orbit by experimental data.

Analog Electronic Implementation of Unstable Dissipative Systems of Type I with Multi-Scrolls Displaced Along Space

Multi-scroll Unstable Dissipative Systems (UDS) in [Formula: see text] which consist of piecewise linear systems are implemented electronically by means of analog computing. The scrolling behavior of the systems can be designed to oscillate along a specific axis or into space depending on the unstable and stable manifolds. In order for a multi-scroll attractor, this switching system must present at least two unstable hyperbolic focus-saddle equilibria with the same stability index, a negative real eigenvalue and a pair of complex conjugated eigenvalues with positive real part. Then, to displace the scrolls among the axes and space different switching control laws must be designed. By taking into consideration the mathematical expressions of the switching systems, the electronic implementations are carried out by means of operational amplifiers representing the real analog physical solution of the systems, from which the voltage is measured representing the states solution.