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Concept# Structural stability

Résumé

In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact C1-small perturbations).
Examples of such qualitative properties are numbers of fixed points and periodic orbits (but not their periods). Unlike Lyapunov stability, which considers perturbations of initial conditions for a fixed system, structural stability deals with perturbations of the system itself. Variants of this notion apply to systems of ordinary differential equations, vector fields on smooth manifolds and flows generated by them, and diffeomorphisms.
Structurally stable systems were introduced by Aleksandr Andronov and Lev Pontryagin in 1937 under the name "systèmes grossiers", or rough systems. They announced a characterization of rough systems in the plane, the Andronov–Pontryagin criterion. In this case, structurally stable systems are typical, they form an open dense set in the space of all systems endowed with appropriate topology. In higher dimensions, this is no longer true, indicating that typical dynamics can be very complex (cf. strange attractor). An important class of structurally stable systems in arbitrary dimensions is given by Anosov diffeomorphisms and flows. During the late 1950s and the early 1960s, Maurício Peixoto and Marília Chaves Peixoto, motivated by the work of Andronov and Pontryagin, developed and proved Peixoto's theorem, the first global characterization of structural stability.
Let G be an open domain in Rn with compact closure and smooth (n−1)-dimensional boundary. Consider the space X1(G) consisting of restrictions to G of C1 vector fields on Rn that are transversal to the boundary of G and are inward oriented. This space is endowed with the C1 metric in the usual fashion. A vector field F ∈ X1(G) is weakly structurally stable if for any sufficiently small perturbation F1, the corresponding flows are topologically equivalent on G: there exists a homeomorphism h: G → G which transforms the oriented trajectories of F into the oriented trajectories of F1.

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Cours associés (8)

Publications associées (4)

Concepts associés (8)

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Structural stability

In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact C1-small perturbations). Examples of such qualitative properties are numbers of fixed points and periodic orbits (but not their periods). Unlike Lyapunov stability, which considers perturbations of initial conditions for a fixed system, structural stability deals with perturbations of the system itself.

Théorie de la stabilité

En mathématiques, la théorie de la stabilité traite la stabilité des solutions d'équations différentielles et des trajectoires des systèmes dynamiques sous des petites perturbations des conditions initiales. L'équation de la chaleur, par exemple, est une équation aux dérivées partielles stable parce que des petites perturbations des conditions initiales conduisent à des faibles variations de la température à un temps ultérieur en raison du principe du maximum.

Cycle limite

Dans le domaine des systèmes dynamiques, un cycle limite, ou cycle-limite sur un plan ou une variété bidimensionnelle est une trajectoire fermée dans l'espace des phases, telle qu'au moins une autre trajectoire spirale à l'intérieur lorsque le temps tend vers . Ces comportements s'observent dans certains systèmes non linéaires. Si toutes les trajectoires voisines approchent le cycle limite lorsque t , on parle de cycle limite stable ou attractif. Si en revanche cela se produit lorsque t , on parle de cycle limite instable ou non attractif.

Séances de cours associées (63)

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An assembly refers to a collection of parts joined together to achieve a specific form and/or functionality. Assemblies make it possible to fabricate large and complex objects with several small and simple parts. Such parts can be assembled and disassembled repeatedly, benefiting the transportation and maintenance of the assembly. Due to these advantages, assemblies are ubiquitous in our daily lives, including most furniture, household appliances, and architecture. The recent advancement in digital fabrication lowers the hurdles for fabricating objects with complex shapes. However, designing physically plausible assemblies is still a non-trivial task as a slight local modification on a part's geometry could have a global impact on the structural and/or functional performance of the whole assembly. New computational tools are developed to enable general users involved in the design process exploiting their imagination. This thesis focuses on static assemblies with rigid parts. We develop computational methods for analyzing and designing assemblies that are structurally stable and assemblable. To address this problem, we use integral joints i.e., tenon and mortise, that are historically used because of their reversibility which simplifies the disassembly process significantly. Properly arranged integral joints can restrict parts' relative movement for improved structural stability. However, manually finding the right joints' geometry is a tedious and error-prone task. Inspired by the kinematic-static duality, we first propose a new kinematic-based method for analyzing the structural stability of assemblies. We then develop a two-stage computational design framework based on this new analyzing method. The kinematic design stage determines the amount of motion restrictions imposed by joints to make a given assembly stable in the motion space. The geometric design stage searches for proper shapes of the joints to satisfy the motion restriction requirements computed from the previous stage. To solve the problem numerically, we propose the joint motion cones to measure the motion restriction capacity of given joints. Compared with previous works, our framework can efficiently handle inputs with complex geometry. Besides, our design framework is very flexible and can easily be adapted to various applications:First, we focus on designing globally interlocking assemblies that can withstand arbitrary external forces and torques. Second, we are interested in designing assemblies of rigid convex blocks to approximate freeform surfaces. Our design framework can optimize the blocks' shape to generate assemblies with good resistance against lateral forces, and in some cases, globally interlocking assemblies.Lastly, we present a method for designing complex assemblies with cone joints. By optimizing the shapes of cone joints, our design framework can find the best trade-off between structural stability and assemblability.We validate our computational tools by fabricating a series of physical prototypes. Our algorithms have great potential to be applied for solving various assembly design problems ranging from small-scale such as toys and furniture to large-scale such as art installation and architecture. For example, the proposed techniques could be applied for designing discrete architecture that can be automatically constructed with robots.

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