Dimensional Stable Manifolds Research Articles

Bifurcation in a Three Dimensional Continuous Fermentation Model

A three dimensional continuous fermentation model with variable yields is proposed in this paper. The properties of the equilibrium points, the global stability, the existence of limit cycles and the Hopf bifurcation in the two dimensional stable manifold of one microorganism while the other is going to vanish in the competition are investigated by qualitative analysis of differential equations.

Connecting Orbits for Compact Infinite Dimensional Maps: Computer Assisted Proofs of Existence

We develop and implement a computer assisted argument for proving the existence of heteroclinic/homoclinic connecting orbits for compact infinite dimensional maps. The argument is based on a posteriori analysis of a certain “discrete time boundary value problem,” and a key ingredient is representing the local stable/unstable manifolds of the fixed points. For a compact mapping the stable manifold is infinite dimensional, and an important component of the present work is the development of computer assisted error bounds for numerical approximation of infinite dimensional stable manifolds. As an illustration of the utility of our method we prove the existence of some connecting orbits for a nonlinear dynamical system which appears in mathematical ecology as a model of a spatially distributed ecosystem with population dispersion.

Counting Critical Formations on a Line

Formation shape control for a collection of point agents is concerned with devising decentralized control laws which ensure that the formation will move so that certain interagent distances approximate prescribed values as closely as possible. Such laws are often derived using steepest descent of a potential function which is invariant under translation and rotation, and then critical formations are those that are fixed under the evolution of the decentralized control dynamics, i.e., those corresponding to equilibrium points of the control dynamics. Using a specific and frequently used potential function for formation control, this paper introduces tools from Morse theory and complex algebraic geometry to estimate the number of critical formations of $N$ agents on a line. We show that there are at least $2N-1$ equilibrium points and at most $3^{N-1}$ isolated equilibria. Moreover, bounds on the number of equilibrium points with a $k$-dimensional stable manifold (the so-called Morse-index) are established....

R&D Spillovers in an Endogenous Growth Model with Physical Capital, Human Capital and Varieties

There is a family of models with Physical, Human capital and RD Gomez, 2005). However, spillovers in R&D have been ignored in this context. We introduce spillovers in this model and derive its steady-state and stability properties. This new feature implies that the model is characterized by a system of four differential equations. A unique Balanced Growth Path along with a two dimensional stable manifold are obtained under simple and reasonable conditions. Transition is oscillatory toward the steady-state for plausible values of parameters.

Equilibrium points of the tilted perfect fluid Bianchi VIh state space

We present the full set of evolution equations for the spatially homogeneous cosmologies of type VI$_h$ filled with a tilted perfect fluid and we provide the corresponding equilibrium points of the resulting dynamical state space. It is found that only when the group parameter satisfies $h>-1$ a self-similar solution exists. In particular we show that for $h>-{\frac 19}$ there exists a self-similar equilibrium point provided that $\gamma \in ({\frac{2(3+\sqrt{-h})}{5+3\sqrt{-h}}},{\frac 32}) $ whereas for $h<-{\frac 19}$ the state parameter belongs to the interval $\gamma \in (1,{\frac{2(3+\sqrt{-h})}{5+3\sqrt{-h}}}) $. This family of new exact self-similar solutions belongs to the subclass $n_\alpha ^\alpha =0$ having non-zero vorticity. In both cases the equilibrium points have a five dimensional stable manifold and may act as future attractors at least for the models satisfying $n_\alpha ^\alpha =0$. Also we give the exact form of the self-similar metrics in terms of the state and group parameters. As an illustrative example we provide the explicit form of the corresponding self-similar radiation model ($\gamma={\frac 43}$), parametrised by the group parameter $h$. Finally we show that there are no tilted self-similar models of type III and irrotational models of type VI$_h$.

Letter: On Tilted Perfect Fluid Bianchi Type VI0Self-Similar Models

We show that the tilted perfect fluid Bianchi VI$_0$ family of self-similar models found by Rosquist and Jantzen [K. Rosquist and R. T. Jantzen, \emph{% Exact power law solutions of the Einstein equations}, 1985 Phys. Lett. \textbf{107}A 29-32] is the most general class of tilted self-similar models but the state parameter $\gamma $ lies in the interval $(\frac 65,\frac 32) $. The model has a four dimensional stable manifold indicating the possibility that it may be future attractor, at least for the subclass of tilted Bianchi VI$_0$ models satisfying $n_\alpha ^\alpha =0$ in which it belongs. In addition the angle of tilt is asymptotically significant at late times suggesting that for the above subclasses of models the tilt is asymptotically extreme.

Numerical Approximation of ( n − 1)-Dimensional Stable Manifolds in Large Systems such as the Power System

This paper discusses a direct method for numerical computation of the local quadratic approximations of ( n − 1)-dimensional stable manifolds for the unstable equilibria on the stability boundary in large systems. The algorithm proposed here does not require time-consuming normal form or symbolic computations. The numerical procedure is illustrated on transient stability problems in simple power system models. © 1997 Elsevier Science Ltd.

Numerical Approximation of (N - 1)-Dimensional Stable Manifolds in Large Systems Such as the Power System

This paper discusses a direct method for the numerical computation of local quadratic approximations of (n-1) dimensional stable manifolds for the unstable equilibria on the stability boundary in large systems. The algorithm proposed here docs not require time-consuming normal form or symbolic computations. The numerical procedure is illustrated on transient stability problems in simple power system models.