Set Of Initial Points Research Articles

Analysis of nonlinear reactor systems by forward and backward integration methods

In this paper, third-order nonlinear nuclear reactor systems are analyzed by plotting trajectories. The unique approach is that an accurate boundary of the asymptotic stability region can be found by first linearizing a system at an unstable singular point, such as a saddle point, and then plotting the trajectories starting from a choosing set of initial points very near the unstable singular point and on the surface spanned by the stable right eigenvector of the linearized system. In addition, the system characteristics, such as speed of response and damping, can be studied by plotting trajectories starting from a preselected sets of initial conditions. Both the forward and backward integration methods are used to calculate the trajectories with a computer. Two reactor systems are analyzed, and comparisons with the other methods in current literature are made.

Indecomposable continua in dynamical systems with noise: Fluid flow past an array of cylinders.

Standard dynamical systems theory is based on the study of invariant sets. However, when noise is added, there are no bounded invariant sets. Our goal is then to study the fractal structure that exists even with noise. The problem we investigate is fluid flow past an array of cylinders. We study a parameter range for which there is a periodic oscillation of the fluid, represented by vortices being shed past each cylinder. Since the motion is periodic in time, we can study a time-1 Poincare map. Then we add a small amount of noise, so that on each iteration the Poincare map is perturbed smoothly, but differently for each time cycle. Fix an x coordinate x(0) and an initial time t(0). We discuss when the set of initial points at a time t(0) whose trajectory (x(t),y(t)) is semibounded (i.e., x(t)>x(0) for all time) has a fractal structure called an indecomposable continuum. We believe that the indecomposable continuum will become a fundamental object in the study of dynamical systems with noise. (c) 1997 American Institute of Physics.

Optimal Control of the Blowup Time

The problem of optimal control of the blowup time of a system of nonlinear controlled ordinary differential equations is considered in this paper. The blowup time is defined to be the first time that the norm of the trajectory becomes infinite. When one seeks to maximize the blowup time the pair $(V(x),\Omega )$ comes under consideration, where $x \in R^n \mapsto V(x) \in [0,\infty ]$ is the value function and $\Omega \subset R^n $ is the blowup set. This is the set of initial points from which finite time blowup will occur for any control. We prove that $(V,\Omega )$ is the unique viscosity solution of the equation $1 + \max _z D_x V(x) \cdot f(x,z) = 0$, $x \in \Omega $ and conditions $\lim _{| x | \to \infty } V(x) = 0$, $\lim _{x \to \partial \Omega } V(x) = + \infty $. Finally, we derive the Pontryagin maximum principle for an optimal control. Some generalizations are also discussed.

Fractal image compression based on Delaunay triangulation and vector quantization

Presents a new scheme for fractal image compression based on adaptive Delaunay triangulation. Such a partition is computed on an initial set of points obtained with a split and merge algorithm in a grey level dependent way. The triangulation is thus fully flexible and returns a limited number of blocks allowing good compression ratios. Moreover, a second original approach is the integration of a classification step based on a modified version of the Lloyd algorithm (vector quantization) in order to reduce the encoding complexity. The vector quantization algorithm is implemented on pixel histograms directly generated from the triangulation. The aim is to reduce the number of comparisons between the two sets of blocks involved in fractal image compression by keeping only the best representative triangles in the domain blocks set. Quality coding results are achieved at rates between 0.25-0.5 b/pixel depending on the nature of the original image and on the number of triangles retained.

Optimal spline fitting to planar shape

Abstract Parametric spline models are used extensively in representing and coding planar curves. For many applications, it is desirable to be able to derive the spline representation from a set of sample joints of the planar shape. The problem we address in this paper is to find a cubic spline model to optimally approximate a given planar shape. We solve this problem by treating the control points which define the spline as variables and apply an optimization technique to minimize an error norm so as to find the best locations of the control points. The error norm, which is defined as the total squared distance of the curve sample points from the spline model, reflects the discrepancy between the spline and the original curve. The objective function for the optimization process is the error norm plus a term which ensures convergence to the correct solution. The initial locations of the control points are selected heuristically. We also describe an extension of this method, which allows the addition of control points to the spline model based on local error measures if the initial set of control points fails to represent the given shape within a prespecified tolerance.

Null-controllability of discrete-time positive control systems

Abstract The discrete-time linear positive and quasilinear monotone systems are considered. The set of initial points is restricted to be the cone of non-negative vectors. The input belongs to the cone of non-positive vectors. The necessary and sufficient conditions of null-controllability for linear systems are obtained. The conditions are sufficient for quasilinear systems.

Parametric piecewise surfaces intersection

A surface-surface intersection method has been developed for free-form parametric surfaces with C 1 continuity. An initial set of intersection points is calculated by any kind of iterative method such as Timmer's[1]. Additional points forming spans of the intersection curve(s) are calculated by an incremental method in the parametric space. The intersection curve is defined as a list of points. The final representation for the curve is user defined.

Newton’s method and symbolic dynamics

By the use of symbolic dynamics, this note proves a result of B. Barna concerning real polynomials of degree at least 4 and having all distinct simple real roots. Specifically, the set of initial points, for which Newton's method fails to converge to a root of the given polynomial, is homeomorphic to a Cantor set. Also the note shows that the requirement for simple roots may be relaxed, and one still has Barna's result being valid.

On strictly positively invariant cones

For a matrix A ∈ R n × n , it is shown that strict positive invariance of a proper cone C ⊂ R n (that is, e tA C/{0} ⊂ int C ∀ t 0) implies the existence of a certain direct sum decomposition of R n into A-invariant subspaces. Our results lead to a characterization of the set of initial points which give rise to solution curves that reach S , under the differential equation ẋ = Ax. Also given is an application in stability theory.