The Construction of Finite Difference Approximations to Ordinary Differential Equations

Finite difference approximations of the form Σ^(si)_(i=-rj)d_(j,i)u_(j+i)=Σ^(mj)_(i=1) e_(j,if)(z_(j,i)) for the numerical solution of linear nth order ordinary differential equations are analyzed. The order of these approximations is shown to be at least r_j + s_j + m_j - n, and higher for certain special choices of the points Z_(j,i). Similar approximations to initial or boundary conditions are also considered and the stability of the resulting schemes is investigated

Global invariant manifolds in the transition to preturbulence in the Lorenz system

AbstractWe consider the homoclinic bifurcation of the Lorenz system, where two primary periodic orbits of saddle type bifurcate from a symmetric pair of homoclinic loops. The two secondary equilibria of the Lorenz system remain the only attractors before and after this bifurcation, but a chaotic saddle is created in a tubular neighbourhood of the two homoclinic loops. This invariant hyperbolic set gives rise to preturbulence, which is characterised by the presence of arbitrarily long transients.In this paper, we show how and where preturbulence arises in the three-dimensional phase space. To this end, we consider how the relevant two-dimensional invariant manifolds — the stable manifolds of the origin and of the primary periodic orbits — organise the phase space of the Lorenz system. More specifically, by means of recently developed and very robust numerical methods, we study how these manifolds intersect a suitable sphere in phase space. In this way, we show how the basins of attraction of the two attracting equilibria change topologically in the homoclinic bifurcation. More specifically, we characterise preturbulence in terms of the accessible boundary between the two basins, which accumulate on each other in a Cantor structure

Nonlinear Numerics

The objectives and some basic methods of numerical bifurcation analysis are described. Several computational examples are used to illustrate the power as well as the limitations of these techniques. Future directions of algorithmic and software development are also discussed. Contents 1 Introduction 4 2 Continuation 4 2.1 Regular solution points : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 Simple singular points : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.3 Linear algebra : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 3 Numerical Bifurcation Analysis of ODEs 8 3.1 Boundary value problems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 3.2 Periodic solutions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 3.3 Connecting orbits : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 3.4 Discretization : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :..

AUTO94P: An Experimental Parallel Version of AUTO

A detailed description is given of the parallel algorithms used in AUTO94P, an experimental parallel version of the software AUTO for the numerical bifurcation analysis of systems of ordinary differential equations. Timing results and user instructions for the Intel Delta are included. The sequential version of the software, AUTO94, is fully described in [8]. For a related tutorial paper see [5, 6]. 1 Introduction In this report we give a detailed presentation of the parallel algorithms used in AUTO94P, an experimental version of the software package AUTO for the bifurcation analysis of systems of ordinary differential equations. The latest version of the standard sequential software, AUTO94, is described in [8], where user instructions and many illustrative examples are given. A description of the numerical algorithms used in AUTO, as well as related algorithms, can be found in [5, 6]. To obtain a copy of AUTO94 or for information on AUTO94P send email to [email protected]. We..

Periodic orbits and synchronous chaos in lasers unidirectionally coupled via saturable absorbers

We study a model for two unidirectionally coupled molecular lasers with a saturable absorber. Our numerical bifurcation study shows the existence of isolas of in-phase periodic solutions as physical parameters change. There are also other non-isola in-phase and intermediate-phase families of periodic oscillations. The coupling parameter strongly affects the stability of these periodic solutions. In this model the unstable periodic orbits belonging to the in-phase isolas constitute a skeleton of the attractor, when chaotic synchronization sets in for a set of physically relevant control parameters

Phase locked periodic solutions and synchronous chaos in a model of two coupled molecular lasers

We study a rate-equation model for two coupled molecular lasers with a saturable absorber. A numerical bifurcation study shows the existence of isolas for in-phase periodic solutions as physical parameters change. In addition there are other non-isola families of in-phase, anti-phase and intermediate-phase periodic oscillations. In this model the unstable periodic orbits belonging to the in-phase isolas constitute a skeleton of the attractor, when chaotic synchronization sets in for a set of physically relevant control parameters