Study of Singularities in Nonsmooth Dynamical Systems via Singular Perturbation

In this article we describe some qualitative and geometric aspects of nonsmooth dynamical systems theory around typical singularities. We also establish an interaction between nonsmooth systems and geometric singular perturbation theory. Such systems are represented by discontinuous vector fields on R(l), l >= 2, where their discontinuity set is a codimension one algebraic variety. By means of a regularization process proceeded by a blow-up technique we are able to bring about some results that bridge the space between discontinuous systems and singularly perturbed smooth systems. We also present an analysis of a subclass of discontinuous vector fields that present transient behavior in the 2-dimensional case, and we dedicate a section to providing sufficient conditions in order for our systems to have local asymptotic stability.

THREE TIME SCALE SINGULAR PERTURBATION PROBLEMS AND NONSMOOTH DYNAMICAL SYSTEMS

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Geometric singular perturbartion theory for non-smooth dynamical systems

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Essential spectrum of systems of singular differential equations

In this paper we develop a new method to determine the essential spectrum of coupled systems of singular differential equations. Applications to problems from magnetohydrodynamics and astrophysics are given.

An optimization method for conceptual design in engineering systems based on high order singular value decomposition

Se presenta un nuevo método de diseño conceptual en Ingeniería Aeronáutica basado el uso de modelos reducidos, también llamados modelos sustitutos (‘surrogates’). Los ingredientes de la función objetivo se calculan para cada indiviudo mediante la utilización de modelos sustitutos asociados a las distintas disciplinas técnicas que se construyen mediante definiciones de descomposición en valores singulares de alto orden (HOSVD) e interpolaciones unidimensionales. Estos modelos sustitutos se obtienen a partir de un número limitado de cálculos CFD. Los modelos sustitutos pueden combinarse, bien con un método de optimización global de tipo algoritmo genético, o con un método local de tipo gradiente. El método resultate es flexible a la par que mucho más eficiente, computacionalmente hablando, que los modelos convencionales basados en el cálculo directo de la función objetivo, especialmente si aparecen un gran número de parámetros de diseño y/o de modelado. El método se ilustra considerando una versión simplificada del diseño conceptual de un avión. Abstract An optimization method for conceptual design in Aeronautics is presented that is based on the use of surrogate models. The various ingredients in the target function are calculated for each individual using surrogates of the associated technical disciplines that are constructed via high order singular value decomposition and one dimensional interpolation. These surrogates result from a limited number of CFD calculated snapshots. The surrogates are combined with an optimization method, which can be either a global optimization method such as a genetic algorithm or a local optimization method, such as a gradient-like method. The resulting method is both flexible and much more computationally efficient than the conventional method based on direct calculation of the target function, especially if a large number of free design parameters and/or tunablemodeling parameters are present. The method is illustrated considering a simplified version of the conceptual design of an aircraft empennage.

Use of the singular value decomposition to increase execution and storage efficiency of the Manteuffel algorithm for the solution of nonsymmetric linear systems /

Bibliography: p. 17-18.

A class of singular first order differential equations with applications in reaction-diffusion

Agências Financiadoras: FCT e MIUR

KAM theory for conformally symplectic systems

We present a KAM theory for some dissipative systems (geometrically, these are conformally symplectic systems, i.e. systems that transform a symplectic form into a multiple of itself). For systems with n degrees of freedom depending on n parameters we show that it is possible to find solutions with n-dimensional (Diophantine) frequencies by adjusting the parameters. We do not assume that the system is close to integrable, but we use an a-posteriori format. Our unknowns are a parameterization of the solution and a parameter. We show that if there is a sufficiently approximate solution of the invariance equation, which also satisfies some explicit non–degeneracy conditions, then there is a true solution nearby. We present results both in Sobolev norms and in analytic norms. The a–posteriori format has several consequences: A) smooth dependence on the parameters, including the singular limit of zero dissipation; B) estimates on the measure of parameters covered by quasi–periodic solutions; C) convergence of perturbative expansions in analytic systems; D) bootstrap of regularity (i.e., that all tori which are smooth enough are analytic if the map is analytic); E) a numerically efficient criterion for the break–down of the quasi–periodic solutions. The proof is based on an iterative quadratically convergent method and on suitable estimates on the (analytical and Sobolev) norms of the approximate solution. The iterative step takes advantage of some geometric identities, which give a very useful coordinate system in the neighborhood of invariant (or approximately invariant) tori. This system of coordinates has several other uses: A) it shows that for dissipative conformally symplectic systems the quasi–periodic solutions are attractors, B) it leads to efficient algorithms, which have been implemented elsewhere. Details of the proof are given mainly for maps, but we also explain the slight modifications needed for flows and we devote the appendix to present explicit algorithms for flows.

Poincar-Cartan integral invariant and canonical transformation for singular lagrangians

In this work we develop the canonical formalism for constrained systems with a finite number of degrees of freedom by making use of the PoincarCartan integral invariant method. A set of variables suitable for the reduction to the physical ones can be obtained by means of a canonical transformation. From the invariance of the PoincarCartan integral under canonical transformations we get the form of the equations of motion for the physical variables of the system.

Canonical Noether symmetries and commutativity properties for gauge systems

For a dynamical system defined by a singular Lagrangian, canonical Noether symmetries are characterized in terms of their commutation relations with the evolution operators of Lagrangian and Hamiltonian formalisms. Separate characterizations are given in phase space, in velocity space, and through an evolution operator that links both spaces. 2000 American Institute of Physics.

Poincar-Cartan intregral invariant and canonical trasformation for singular Lagrangians: an addendum

The results of a previous work, concerning a method for performing the canonical formalism for constrained systems, are extended when the canonical transformation proposed in that paper is explicitly time dependent.

Coisotropic regularization of singular Lagrangians

We present an alternative approach to the usual treatments of singular Lagrangians. It is based on a Hamiltonian regularization scheme inspired on the coisotropic embedding of presymplectic systems. A Lagrangian regularization of a singular Lagrangian is a regular Lagrangian defined on an extended velocity phase space that reproduces the original theory when restricted to the initial configuration space. A Lagrangian regularization does not always exists, but a family of singular Lagrangians is studied for which such a regularization can be described explicitly. These regularizations turn out to be essentially unique and provide an alternative setting to quantize the corresponding physical systems. These ideas can be applied both in classical mechanics and field theories. Several examples are discussed in detail. 1995 American Institute of Physics.

Robust power allocation algorithms for MIMO OFDM systems with imperfect CSI

This paper presents a Bayesian approach to the design of transmit prefiltering matrices in closed-loop schemes robust to channel estimation errors. The algorithms are derived for a multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) system. Two different optimizationcriteria are analyzed: the minimization of the mean square error and the minimization of the bit error rate. In both cases, the transmitter design is based on the singular value decomposition (SVD) of the conditional mean of the channel response, given the channel estimate. The performance of the proposed algorithms is analyzed,and their relationship with existing algorithms is indicated. As withother previously proposed solutions, the minimum bit error rate algorithmconverges to the open-loop transmission scheme for very poor CSI estimates.