Note on Logarithmic Switchback Terms in Regular and Singular Perturbation Expansions

Abstract

The occurrence of logarithmic switchback is studied for ordinary differential equations containing a parameter k which is allowed to take any value in a continuum of real numbers and with boundary conditions imposed at $x = \varepsilon $ and $x = \infty $. Classical theory tells us that if the equation has a regular singular point at the origin there is a family of solutions which varies continuously with k, and the expansion around the origin has log x terms for a discrete set of values of k. It is shown here how nonlinearity enlarges this set so that it may even be dense in some interval of the real numbers. A $\log x$ term in the expansion in x leads to expansion coefficients containing $\log\varepsilon $ (switchback) in the perturbation expansion. If for a given value of k logarithmic terms in x and $\varepsilon $ occur they may be obtained by continuity from neighboring values of k. Switchback terms occurred conspicuously in singular-perturbation solutions of problems posed for semi-infinite domain $x\geqq \varepsilon $. This connection is historical rather than logical. In particular we study here switchback terms for a specific example using methods of both singular and regular perturbations.