Concept Of Manifold Research Articles

Noncontinuous Minkowskian spacetime

A model for a noncontinuous spacetime is considered, which is based on a generalization of the manifold concept, known as d-space. As a consequence of its construction, the model possesses a metric of Lorentzian signature and a generalized form of Lorentz invariance. The model may be considered as defining a class of discrete spacetimes within the framework of the d-space representation of general relativity.

A learning scheme for dynamic neural networks: Equilibrium manifold and connective stability

Stability of recurrent neural network N: x ̇ i =x i + ∑ j=1 n e ij w ij 8 j (x i )+I i , i=1,2,…,n y i =g i (x i ) with learning rule L: e ̇ ij =uh ij (e,y,y ∗ ), i,j=1,2,…,n is analyzed using the concept of equilibrium manifold. We shall show that the concept provides an ideal setting for the formulation of a learning rule L that adaptively teaches network N to acquire a desired pattern y ∗ as one of its asymptotically stable equilibria. Connective stability of a moving equilibrium in the composite two-time-scale system N & L is established for bounded interconnection weights e ij w ij and a sufficiently small learning rate μ. The conditions for connective stability, which are derived using the M-matrices and the concept of vector Liapunov functions, are suitable for studying the design trade-offs between the bounds on the nominal weights w ij , the shape of the sigmoid function g i (x i ), the learning rule h (e, y, y ∗ ), and rate μ, as well as the size of the stability region containing y ∗ .

The structure of the b-completion of space-time

Structured space, as a natural generalization of the manifold concept, is defined to be a topological space with a sheaf of real function algebras which are suitably localized and closed with respect to composition with smooth Euclidean functions. Vector fields, differential forms, linear connection and curvature are introduced on structured spaces. It is shown that structured spaces correctly model space-times with singularities. Schmidt's b-boundary of space-time is constructed in the category of structured spaces, and well known difficulties with the b-boundaries of the closed Friedman and Schwarzschild space-times are disentangled. It is argued that the b-boundary of space-time, when considered in the category of structured spaces, can serve as a good definition of classical singularities.

The squeezing property and the decay of small wavelengths in dissipative dynamical systems

The concepts of inertial manifolds and approximate inertial manifolds for dissipative dynamical systems are based upon a decomposition of the underlying phase space into a finitedimensional, inertial subspace, and an infinite-dimensional,highly dissipated, orthogonal complement. We present here a more detailed squeezing property which shows that, under the same conditions as the usual queering property, there exists a small parameter r > 0 such that the percentage p of the total energy contained in the highly dissipated modes is either dose to unity, and the entire system is rapidly dissipatingto rero, or p passes a threshold value 1 e end subsequently decays exponentially with a large rate to a value inferior to E, and remains there for dl forward time.

How Hyperbolic Geometry Became Respectable

(1994). How Hyperbolic Geometry Became Respectable. The American Mathematical Monthly: Vol. 101, No. 5, pp. 464-470.

Numerical calculation of invariant manifolds for maps

AbstractMany processes in the sciences and in engineering are modelled by dynamical systems and—in discretized version—by nonlinear maps. To understand the often complicated dynamical behaviour it is a well established tool to use the concept of invariant manifolds of the system. In this way it is often possible to reduce the dimension of the system considerably. In this paper we propose a new method to calculate numerically invariant manifolds near fixed points of maps. We prove convergence of our procedure and provide an error estimation. Finally, the application of the method is illustrated by examples.

Full discrete nonlinear galerkin method for the Navier-Stokes equations

This paper deals with the inertial manifold and the approximate inertial manifold concepts of the Navier-Stokes equations with nonhomogeneous boundary conditions and inertial algorithm. Furtheremore, we provide the error estimates of the approximate solutions of the Navier-Stokes Equations.

On hypoellipticity and solvability of linear partial differential operators with multiple characteristics

The concept of characteristic manifold is very important in PDE, but it takes into account only the principal symbol. This paper is one of several papers devoted to the following problem: how can we construct the “characteristic manifold” that takes into account lower order terms and at the same time describes the intrinsic properties of the operator (like hypoellipticity or solvability) in the same way as the classical characteristic manifold does.

Robust adaptive controller design and stability analysis for flexible-joint manipulators

The problem of controlling robot manipulators with flexible joints is considered. A reduced-order flexible-joint model based on a singular perturbation formulation of the manipulator equations of motion is used. The concept of an integral manifold is utilized to construct the dynamics of the slow subsystem. A fast subsystem is constructed to represent the dynamics of the elastic forces at the joints. A composite adaptive control scheme is developed with special attention to stability and robustness of the controller. The proposed controller is based on online identification of the manipulator parameters and takes into account the effect of a class of unmodeled dynamics, identification errors, and parameter variations. Stability analysis of the resulting closed-loop full-order system is presented. To show the capability of the proposed algorithm, an example of a two-link flexible-joint manipulator is considered. Simulation results are given to illustrate the applicability of the proposed control scheme. >

On the Control of Singularly Perturbed Nonlinear Systems

Using the integral manifold concept, a recent systematic method to control ε-perturbed nonlinear systems is here extended to a rather wide class of singularly perturbed MIMO systems. This class includes singularly perturbed systems exibiting an affine structure in the control after order reduction. It is shown that approximated input-output linearization results at any desired order can be achieved by means of successive diffeomorphisms and corrective control laws, based on the reduced-order slow system. Some simulation results illustrate the efficiency of the proposed control strategy.

Robust adaptive controller design for flexible joint manipulators

Abstract In this paper the control problem for robot manipulators with flexible joints is considered. A reduced-order flexible joint model is constructed based on a singular perturbation formulation of the manipulator equations of motion. The concept of an integral manifold is utilized to construct the dynamics of a slow subsystem. A fast subsystem is constructed to represent the fast dynamics of the elastic forces at the joints. A composite control scheme is developed based on on-line identification of the manipulator parameters which takes into account the effect of certain unmodeled dynamics and parameter variations. Stability analysis of the resulting closed-loop full-order system is presented. Simulation results for a single link flexible joint manipulator are given to illustrate the applicability of the proposed algorithm.

Integral manifolds of systems of differential equations with a random right-hand side in a Banach space

The concept of an integral manifold of a system of differential equations with a random right-hand side is introduced. The problem of the existence of an integral manifold of a certain class of differential equations in a Banach space and several of its properties are investigated.

Cylindrical manifolds and reactive island kinetic theory in the time domain

In a series of recent publications we discussed the concept of cylindrical manifolds and their role in mediating the reaction dynamics of chemical reactions. The cylindrical manifolds were used to develop a chemical reaction rate theory we called reactive island (RI) theory. RI theory was cast in terms of the map dynamics between n Poincaré mapping planes—which were referred to as the ‘‘n-map.’’ Therefore ‘‘time’’ did not explicitly appear within n-map RI theory. In this paper we extend n-map RI theory to the time domain. The formal theory, cast as a master equation, is used to obtain the temporal RI model. Temporal RI theory is applied to a two degree-of-freedom system exhibiting dynamical chaos. The results of temporal RI theory are compared with classical Rice–Ramsberger–Kassel–Marcus (RRKM) theory and it is found that even under, presumably, ‘‘ideal’’ dynamical conditions, RRKM theory can be in serious error with numerical simulation. It is also seen that temporal RI theory accurately accounts for the rates at energies far above the barrier, where RRKM theory is not expected to be applicable. We also discuss some rigorous comparisons between the decay rates obtained from n-map and temporal RI theories.

Approximation of attractors, large eddy simulations and multiscale methods

Recent advances in the mathematical theory of the Navier-Stokes equations have produced new insight in the mathematical theory of turbulence. In particular, the study of the attractor for the Navier-Stokes equations produced the first connection between two approaches to turbulence that seemed far apart, namely the conventional approach of Kolmogorov and the dynamical systems theory approach. Similarly the study of the approximation of the attractor in connection with the newly introduced concept of approximate inertial manifolds has produced a new approach to large eddy simulations and the study of the interaction of small and large eddies in turbulent flows. Our aim in this article is to survey and describe some of the new results concerning the functional properties of the Navier-Stokes equations and to discuss their relevance to turbulence.

Chaotic Motions of a Constrained Pipe Conveying Fluid: Comparison Between Simulation, Analysis, and Experiment

A refined analytical model is presented for the dynamics of a cantilevered pipe conveying fluid and constrained by motion limiting restraints. Calculations with the discretized form of this model with a progressively increasing number of degrees of freedom, N, show that convergence is achieved with N = 4 or 5, which agrees with previously performed fractal dimension calculations of experimental data. Theory shows that, beyond the Hopf bifurcation, as the flow is increased, a pitchfork bifurcation is followed by a cascade of period doubling bifurcations leading to chaos, which is in qualitative agreement with observation. The numerically computed theoretical critical flow velocities are in excellent quantitative agreement (5–10 percent) with experimental values for the thresholds of the Hopf and period doubling bifurcations and for the onset of chaos. An approximation for the critical flow velocity for the loss of stability of the post-Hopf limit cycle is also obtained by using center manifold concepts and normal form techniques for a simplified version of the analytical model; it is found that the values obtained in this manner are approximately within 10 percent of those computed numerically.

Approximate inertial manifolds and effective viscosity in turbulent flows

The recently formulated concept of approximate inertial manifolds is exploited as a means for eliminating systematically the fine structure of the velocity field in two-dimensional flows. The resulting iterative procedure does not invoke any statistical properties of the solutions of Navier–Stokes equations. It leads to a modification of those equations, such that effective viscosity-like terms arise in a natural way. The rigorous mathematical considerations can be related to the corresponding physical concepts and intuition. The result leading to a numerical algorithm, essentially a nonlinear Galerkin method, provides a basis for large eddy simulation in which the subgrid model is derived from the properties of the Navier–Stokes equations, rather than from more or less justifiable ad hoc arguments. Some limited speculations concerning the expected results in three dimensions are also offered.

Inertial Manifolds and Slow Manifolds

The concept of slow manifold has been introduced in meteorology and is broadly used for short term weather forecast and data assimilation (see e.g.,[1,2]). Our object in this article is to present a mathematical theory of slow manifolds, using the concept of inertial manifold recently studied in dynamical systems theory. We show in fact that a slow manifold is a special type of inertial manifold. Our study relies on a new construction of inertial manifolds for evolution equations of a more general type than those previously studied.

The differential and cone structures of spacetime

We propose to model spacetime by a differential space rather than by a differential manifold. A differential space is the pair (M, C), where M is any set, and C a family of real functions on M, satisfying certain axioms; C is called a differential structure of a corresponding differential space. This concept suitably generalizes the manifold concept. We show that C can be chosen in such a way that it contains all information about the causal structure of spacetime. This information can be read out of C with the help of only one postulate, namely that physical signals travel along piecewise smooth curves in (M, C). We effectively construct the Minkowski spacetime, with its cone structure, in this way. Some comments are made.

Inertial Manifolds and Multigrid Methods

This article presents an analogy existing between the concepts of approximate inertial manifolds in dynamical systems theory and multigrid methods in numerical analysis. In view of the large-time approximation of dissipative evolution equations in a turbulent regime, a new algorithm is proposed and studied that combines some ideas and concepts of inertial manifolds and multigrid methods. This article emphasizes theoretical questions. More practical (computational) questions will be investigated elsewhere.

Approximate inertial manifolds for reaction-diffusion equations in high space dimension

The concept of approximate inertial manifolds was introduced by Foiaset al. (1987) in the case of the two-dimensional Navier-Stokes equations. These manifolds are finite dimensional smooth manifolds such that the orbits enter a very thin neighborhood of the manifold after a transient time; this concept replaces the one of inertial manifold when either an inertial manifold does not exist or its existence is not known. Our aim in this paper is to prove that approximate inertial manifolds exist for reaction-diffusion equations in high space dimension by opposition with exact inertial manifolds whose existence has only been proved in low dimension and for which nonexistence results have been obtained in space dimensionn=4.