Lagerstrom, P. A. and Reinelt, D. A. (1984) Note on Logarithmic Switchback Terms in Regular and Singular Perturbation Expansions. SIAM Journal on Applied Mathematics, 44 (3). pp. 451462. ISSN 00361399. http://resolver.caltech.edu/CaltechAUTHORS:LAGsiamjam84

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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 = ε and x = ∞. 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 ε (switchback) in the perturbation expansion. If for a given value of k logarithmic terms in x and ε occur they may be obtained by continuity from neighboring values of k. Switchback terms occurred conspicuously in singularperturbation solutions of problems posed for semiinfinite domain x ≥ ε. 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.
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Additional Information:  ©1984 Society for Industrial and Applied Mathematics. Received by the editors August 18, 1983. This author [P.A.L.] gratefully acknowledges the help and hospitality of the Fondation des Treilles. 
Record Number:  CaltechAUTHORS:LAGsiamjam84 
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:LAGsiamjam84 
Alternative URL:  http://dx.doi.org/10.1137/0144030 
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