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Discrete & Continuous Dynamical Systems - S

October 2019 , Volume 12 , Issue 6

Issue dedicated to Alain Léger

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Preface
Laurence Cherfils, Alain Cimetière, Aziz Hamdouni, Michel Jean and Elaine Pratt
2019, 12(6): ⅰ-ⅱ doi: 10.3934/dcdss.201906i +[Abstract](1029) +[HTML](110) +[PDF](81.11KB)
Abstract:
Existence theorem for a first-order Koiter nonlinear shell model
Sylvia Anicic
2019, 12(6): 1535-1545 doi: 10.3934/dcdss.2019106 +[Abstract](1635) +[HTML](390) +[PDF](321.52KB)
Abstract:

We prove the existence of a minimizer for a nonlinearly elastic shell model which coincides to within the first order with respect to small thickness and change of metric and curvature energies with the Koiter nonlinear shell model.

Formal asymptotic analysis of elastic beams and thin-walled beams: A derivation of the Vlassov equations and their generalization to the anisotropic heterogeneous case
Patrick Ballard and Bernadette Miara
2019, 12(6): 1547-1588 doi: 10.3934/dcdss.2019107 +[Abstract](1662) +[HTML](391) +[PDF](519.08KB)
Abstract:

The modelling of ordinary beams and thin-walled beams is rigorously obtained from a formal asymptotic analysis of three-dimensional linear elasticity. In the case of isotropic homogeneous elasticity, ordinary beams yield the Navier-Bernoulli beam model, thin-walled beams with open profile yield the Vlassov beam model and thin-walled beams with closed profile the Navier-Bernoulli beam model. The formal asymptotic analysis is also extensively performed in the case of the most general anisotropic transversely heterogeneous material (meaning the heterogeneity is the same in every cross-section), delivering the same qualitative results. We prove, in particular, the non-intuitive fact that the warping function appearing in the Vlassov model for general anisotropic transversely heterogeneous material, is the same as the one appearing in the isotropic homogeneous case. In the general case of anisotropic transversely heterogeneous material, the analysis provides a rigorous and systematic constructive procedure for calculating the reduced elastic moduli, both in Navier-Bernoulli and Vlassov theories.

Magnetic forces in and on a magnet
Alain Bossavit
2019, 12(6): 1589-1600 doi: 10.3934/dcdss.2019108 +[Abstract](1514) +[HTML](427) +[PDF](333.8KB)
Abstract:

Given the shape of a magnet and its magnetization, point by point, which force does it exert on itself, also point by point? We explain what 'force' means in such a context and how to define it by using the Virtual Power Principle. Mathematically speaking, this force is a vector-valued distribution, with Dirac-like concentrations on surfaces across which the magnetization is discontinuous, i.e., material interfaces. To find these concentrations, we express the force as the divergence of a (symmetric) 2-tensor which generalizes a little the classical Maxwell tensor.

Analysis of discretized parabolic problems modeling electrostatic micro-electromechanical systems
Laurence Cherfils, Alain Miranville, Shuiran Peng and Chuanju Xu
2019, 12(6): 1601-1621 doi: 10.3934/dcdss.2019109 +[Abstract](1514) +[HTML](428) +[PDF](884.5KB)
Abstract:

Our aim in this paper is to study discretized parabolic problems modeling electrostatic micro-electromechanical systems (MEMS). In particular, we prove, both for semi-implicit and implicit semi-discrete schemes, that, under proper assumptions, the solutions are monotonically and pointwise convergent to the minimal solution to the corresponding elliptic partial differential equation. We also study the fully discretized semi-implicit scheme in one space dimension. We finally give numerical simulations which illustrate the behavior of the solutions both in one and two space dimensions.

Complex ray in anisotropic solids: Extended Fermat's principle
Marc Deschamps and Olivier Poncelet
2019, 12(6): 1623-1633 doi: 10.3934/dcdss.2019110 +[Abstract](1486) +[HTML](400) +[PDF](795.84KB)
Abstract:

In contrast to homogeneous plane waves, solutions of the Chris-toffel equation for anisotropic media, for which a determined number of rays can be observed in a fixed direction of observation, inhomogeneous plane waves provide a continuum of "rays" that propagate in this direction. From this continuum, some complex plane waves can be extracted for verifying a definition of quasi-arrivals, based on the condition that the time of flight would vary the less in extension to the Fermat's principle that stipulates a stationary time of flight for wave arrivals. The dynamic response in some angular zones contain prominent, although not singular, features whose arrivals cannot be described by the classical ray theory. These wave packet's arrivals can be described by quasi-fronts associated to specific inhomogeneous plane waves. The extent of the phenomena depends on the degree of anisotropy. For weak anisotropy, such quasi-fronts can be observed. For strong anisotropy, the use of inhomogeneous plane waves, due to their complex slowness vector, permits a simple description of quasi-arrivals that refer to the internal diffraction phenomenon. Some examples are given for different wave surfaces, showing how the wave fronts can be extended beyond the cuspidal edges for forming closed wave surfaces.

On the controllability of racing sailing boats with foils
Philippe Destuynder and Caroline Fabre
2019, 12(6): 1635-1668 doi: 10.3934/dcdss.2019111 +[Abstract](1544) +[HTML](374) +[PDF](10229.27KB)
Abstract:

The development of foils for racing boats has changed the strategy of sailing. Recently, the America's cup held in San Francisco, has been the theatre of a tragicomic history due to the foils. During the last round, the New-Zealand boat was winning by 8 to 1 against the defender USA. The winner is the first with 9 victories. USA team understood suddenly (may be) how to use the control of the pitching of the main foils by adjusting the rake in order to stabilize the ship. And USA won by 9 victories against 8 to the challenger NZ. Our goal in this paper is to point out few aspects which could be taken into account in order to improve this mysterious control law which is known as the key of the victory of the USA team. There are certainly many reasons and in particular the cleverness of the sailors and of all the engineering team behind this project. But it appears interesting to have a mathematical discussion, even if it is a partial one, on the mechanical behaviour of these extraordinary sailing boats. The numerical examples given here are not the true ones. They have just been invented in order to explain the theoretical developments concerning three points: the possibility of tacking on the foils for sailing upwind, the nature of foiling instabilities, if there are, when the boat is flying and the control laws.

Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry
Sanjay Dharmavaram and Timothy J. Healey
2019, 12(6): 1669-1684 doi: 10.3934/dcdss.2019112 +[Abstract](1768) +[HTML](412) +[PDF](6179.96KB)
Abstract:

We consider bifurcation problems in the presence of \begin{document}$ O(3) $\end{document} symmetry. Well known group-theoretic techniques enable the classification of all maximal isotropy subgroups of \begin{document}$ O(3) $\end{document}, with associated mode numbers \begin{document}$\ell∈\mathbb{N} $\end{document}, leading to 1-dimensional fixed-point subspaces of the \begin{document}$ (2\ell+1) $\end{document}-dimensional space of spherical harmonics. In each case the so-called equivariant branching lemma can then be used to establish the existence of a local branch of bifurcating solutions having the symmetry of the respective subgroup. To first-order, such a branch is a precise linear combination of the \begin{document}$ 2\ell+1 $\end{document} spherical harmonics, which we call the bifurcation direction. Our work here is focused on the direct construction of these bifurcation directions, complementing the above-mentioned classification. The approach is an application of a general method for constructing families of symmetric spherical harmonics, based on differentiating the fundamental solution of Laplace's equation in \begin{document}$ \mathbb{R}^3 $\end{document}.

Efficient high-order implicit solvers for the dynamic of thin-walled beams with open cross section under external arbitrary loadings
Ahmed El Kaimbillah, Oussama Bourihane, Bouazza Braikat, Mohammad Jamal, Foudil Mohri and Noureddine Damil
2019, 12(6): 1685-1708 doi: 10.3934/dcdss.2019113 +[Abstract](1814) +[HTML](682) +[PDF](3536.02KB)
Abstract:

This paper aims to investigate, in large displacement and torsion context, the nonlinear dynamic behavior of thin-walled beams with open cross section subjected to various loadings by high-order implicit solvers. These homotopy transformations consist to modify the nonlinear discretized dynamic problem by introducing an arbitrary invertible pre-conditioner \begin{document}$ [K^\star] $\end{document} and an arbitrary path following parameter. The nonlinear strongly coupled equations of these structures are derived by using a \begin{document}$ 3D $\end{document} nonlinear dynamic model which accounts for large displacements and large torsion without any assumption on torsion angle amplitude. Coupling complex structural phenomena such that warping, bending-bending, and flexural-torsion are taken into account.

Two examples of great practical interest of nonlinear dynamic problems of various thin-walled beams with open section are presented to validate the efficiency and accuracy of high-order implicit solvers. The obtained results show that the proposed homotopy transformations reveal a few number of matrix triangulations. A comparison with Abaqus code is presented.

Approximation of semi-groups in the sense of Trotter and asymptotic mathematical modeling in physics of continuous media
Christian Licht and Thibaut Weller
2019, 12(6): 1709-1741 doi: 10.3934/dcdss.2019114 +[Abstract](1773) +[HTML](411) +[PDF](544.42KB)
Abstract:

We derive several models in Physics of continuous media using Trotter theory of convergence of semi-groups of operators acting on variable spaces.

POD basis interpolation via Inverse Distance Weighting on Grassmann manifolds
Rolando Mosquera, Aziz Hamdouni, Abdallah El Hamidi and Cyrille Allery
2019, 12(6): 1743-1759 doi: 10.3934/dcdss.2019115 +[Abstract](1826) +[HTML](439) +[PDF](1027.49KB)
Abstract:

An adaptation of the Inverse Distance Weighting (IDW) method to the Grassmann manifold is carried out for interpolation of parametric POD bases. Our approach does not depend on the choice of a reference point on the Grassmann manifold to perform the interpolation, moreover our results are more accurate than those obtained in [7]. In return, our approach is not direct but iterative and its relevance depends on the choice of the weighting functions which are inversely proportional to the distance to the parameter. More judicious choices of such weighting functions can be carried out via kriging technics [23], this is the subject of a work in progress.

Dynamics of locally linearizable complex two dimensional cubic Hamiltonian systems
Yangyou Pan, Yuzhen Bai and Xiang Zhang
2019, 12(6): 1761-1774 doi: 10.3934/dcdss.2019116 +[Abstract](1575) +[HTML](397) +[PDF](377.7KB)
Abstract:

The aim of this paper is to characterize global dynamics of locally linearizable complex two dimensional cubic Hamiltonian systems. By finding invariants, we prove that their associated real phase space \begin{document}$\mathbb R^4$\end{document} is foliated by two dimensional invariant surfaces, which could be either simple connected, or double connected, or triple connected, or quadruple connected. On each of the invariant surfaces all regular orbits are heteroclinic ones, which connect two singularities, either both finite, or one finite and another at infinity, or both at infinity, and all these situations are realizable.

Topological remarks and new examples of persistence of diversity in biological dynamics
Evariste Sanchez-Palencia and Jean-Pierre Françoise
2019, 12(6): 1775-1789 doi: 10.3934/dcdss.2019117 +[Abstract](1535) +[HTML](379) +[PDF](745.0KB)
Abstract:

There are several definitions of persistence of species, which amount to define interactions between them ensuring the survival of all the species initially present in the system. The aim of this paper is to present a wide family of examples in dimension \begin{document}$n>2$\end{document} (very natural in biological dynamics) exhibiting convergence towards a cycle when starting from anywhere with the exception of a zero-measure set of "forbidden" initial positions. The forbidden set is a heteroclinic orbit linking two equilibria on the boundary of the domain. Moreover, such systems have no equilibrium point interior to the domain (which is necessary for classical persistence for topological reasons). Such systems do not enjoy persistence in a strict sense, whereas in practice they do. The forbidden initial set does not matter in practice, but it modifies drastically the topological properties.

Multi-point Taylor series to solve differential equations
Djédjé Sylvain Zézé, Michel Potier-Ferry and Yannick Tampango
2019, 12(6): 1791-1806 doi: 10.3934/dcdss.2019118 +[Abstract](2108) +[HTML](421) +[PDF](455.71KB)
Abstract:

The use of Taylor series is an effective numerical method to solve ordinary differential equations but this fails when the sought function is not analytic or when it has singularities close to the domain. These drawbacks can be partially removed by considering multi-point Taylor series, but up to now there are only few applications of the latter method in the literature and not for problems with very localized solutions. In this respect, a new numerical procedure is presented that works for an arbitrary cloud of expansion points and it is assessed from several numerical experiments.

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