Part I:
We consider folds in the solution surface of nonlinear equations with two free parameters. A system of equations whose solutions are fold paths is formulated and proved to be non-singular in a neighborhood of a fold, thus making continuation possible. Efficient numerical algorithms employing block Gaussian elimination are developed for applying Euler-Newton pseudo-arclength continuation to the system, and these are shown to require fewer operations than other methods.
To demonstrate the use of these methods we calculate the flow between two infinite, rotating disks. For Reynold's number less than 1000, six separate solution sheets are found and completely described. Plots of 47 solutions for three values of the disk speed ratio and for Reynold's number equal to 625 are shown. These are compared with the solutions found by previous investigators.
Part II:
Two ordinary differential equations with parameters whose solution paths exhibit an infinite sequence of folds clustered about a limiting value are studied. Using phase-plane analysis, expressions for the limiting ratios of the parameter values at which these folds occur are derived and the limiting values are shown to be non-universal.
Part III:
A mesh selection algorithm for use in a code to solve first-order nonlinear two-point boundary value problems with separated end conditions is described. The method is based on equidistributing the global error of the box scheme, a numerical estimate of which is obtained from Richardson extrapolation. Details of the algorithm and examples of its performance on non-stiff and stiff problems are presented.
