Effective bifurcation analysis: a time-stepper-based approach

Abstract

We introduce a numerical approach to perform the effective (coarse-scale) bifurcation analysis of solutions of dissipative evolution equations with spatially varying coefficients. The advantage of this approach is that the 'coarse model' (the averaged, effective equation) need not be explicitly constructed. The method only uses a time-integrator code for the detailed problem and judicious choices of initial data and integration times; the bifurcation computations are based on the so-called recursive projection method (Shroff and Keller 1993 SIAM J. Numer. Anal. 30 1099-120).

Reduction and reconstruction for self-similar dynamical systems

“…Second, since the neutral directions of the symmetry-reduced equations are known (up to some discretization error), it is possible to modify the recursive projection method [20] or the Newton-Picard method [9] to compute steady-states or periodic solutions of these equations. If it is impossible to adapt an existing code to integrate the symmetry-reduced equations, it should still be possible to compute and analyse the unknown symmetry-reduced system using the approach of [19], which does not explicitly need the symmetry-reduced equations but relies only on a black-box time integrator, combined with symmetry transformations of the state. Combining this discretetime approach with the coarse integration and bifurcation techniques we have been recently developing [6] may help with computer-assisted analysis of PDE-level, 'coarse' self-similar solutions for problems for which only microscopic or stochastic descriptions are available.…”

“…Factoring out the group invariance (through so-called pinning conditions) is also an important component of many numerical methods for systems with symmetry. For instance, a similar templatebased phase condition is often used in the computation of periodic solutions of autonomous ordinary differential equations (ODEs) [4], and a related method was recently used in [19] for bifurcation analysis of systems where the governing equations are not explicitly known, but only a numerical timestepper is available.…”

On the application of “equation-free modelling” to neural systems

“…We now give a brief introduction to EF modelling; much more detail can be found elsewhere (Gear et al, 2002;Kevrekidis et al, 2003;Makeev et al, 2002;Möller et al, 2005;Runborg et al, 2002). In Sec.…”

“…Equation-free (EF) modelling, developed in the past few years by Kevrekidis et al (Gear et al, 2002;Kevrekidis et al, 2003;Makeev et al, 2002;Möller et al, 2005;Runborg et al, 2002), is a way of analysing the macroscopic equations for such a system, even though the equations are not known explicitly, by using short bursts of appropriatelyinitialised simulations of the microscopic dynamics. Its success relies on there being a separation of time-scales in the system, with the macroscopic variables of interest changing on a much longer time-scale than most of the microscopic variables.…”

Spatially distributed stochastic systems: Equation-free and equation-assisted preconditioned computations

“…It becomes then important to extract useful coarse-grained, macroscopic information from the microscopic molecular-based model using as few detailed simulations as possible. This is the goal of equation-free methods 3,4,5,6,7,8,9,10 which were designed for cases where the exact macroscopic equations are unavailable in closed form.…”