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

Periodic Orbits Associated with the Libration Points of the Homogeneous Rotating Gravitating Triaxial Ellipsoid

We present computational results for the families of periodic orbits that emanate from the five libration points of the homogeneous gravitating triaxial ellipsoid rotating around its small axis, as well as for various secondary bifurcating families. Possible applications of our results include research on the motion of stars and clusters in elliptical galaxies, and the design of space missions to the vicinity of small bodies (asteroids) and their libration points. The numerical continuation and bifurcation algorithms employed in our study are based on boundary value techniques, as implemented in the AUTO software tool

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

Isolas of periodic passive Q-switching self-pulsations in the three-level:two-level model for a laser with a saturable absorber

We show that a fundamental feature of the three-level:two-level model, used to describe molecular monomode lasers with a saturable absorber, is the existence of isolas of periodic passive Q-switching (PQS) self-pulsations. A common feature of these closed families of periodic solutions is that they contain regions of stability of the PQS self-pulsation bordered by period-doubling and fold bifurcations, when the control parameter is either the incoherent external pump or the cavity frequency detuning. These findings unveil the fundamental solution structure that is at the origin of the phenomenon known as “period-adding cascades” in our system. Using numerical continuation techniques we determine these isolas systematically, as well as the changes they undergo as secondary parameters are varied

Hysteresis of periodic and chaotic passive q-switching self-pulsation in a molecular laser model, and the stark effect as a codimension-2 parameter

We give a systematic comparison of a molecular model for a CO2 laser with a fast saturable absorber and a reduced version of this model. Overall, we find that there is good agreement between these models. We use numerical continuation algorithms to analyze the bifurcation structure of the equations, and complement the results by numerical simulations to model possible experiments. Our study predicts the existence of isolas of periodic passive Q-switching self-pulsations and a rich structure of Q-intervals of stability for these periodic orbits, where Q represents the incoherent pump of the laser. These intervals correspond to the observed phenomenon known as period-adding cascades. Computed loci of codimension-1 bifurcations show that a small shift of a secondary parameter in the reduced model with respect to that of the complete model substantially improves their quantitative agreement. This parameter is associated with the action of the Stark effect in the absorber. We also discuss a necessary condition for chaotic windows to arise as Q changes

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 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :..