Bouquet, S. E.; Feix, M. R.; Leach, P. G. L. Properties of second-order ordinary differential equations invariant under time translation and self-similar transformation. (English) Zbl 0734.34029 J. Math. Phys. 32, No. 6, 1480-1490 (1991). Summary: The general form of a second-order ordinary differential equation invariant under time translation and self-similarity is obtained together with its solution in parametric form. Next two representatives of two different families of such equations are studied. In terms of rescaled phase space variables, the first example has a simple physical interpretation and its limit cycle properties and period are easily derived. Similar results are found for the second example, but the physical interpretation is less obvious. The invariant equations are used as a basis to study the asymptotic behavior of related noninvariant equations thereby underlining the critical value nature of the parameters for which a self-similar solution (SSS) exists. Cited in 10 Documents MSC: 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 34D05 Asymptotic properties of solutions to ordinary differential equations 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. 34A34 Nonlinear ordinary differential equations and systems Keywords:second-order ordinary differential equation; time translation; self- similarity; physical interpretation; limit cycle; period; asymptotic behavior PDFBibTeX XMLCite \textit{S. E. Bouquet} et al., J. Math. Phys. 32, No. 6, 1480--1490 (1991; Zbl 0734.34029) Full Text: DOI References: [1] DOI: 10.1063/1.524932 · Zbl 0459.70024 · doi:10.1063/1.524932 [2] Mahomed F. M., J. Math. Anal. App. 198 pp 358– (1990) [3] DOI: 10.1063/1.528511 · Zbl 0724.34003 · doi:10.1063/1.528511 [4] DOI: 10.1088/0305-4470/20/2/014 · Zbl 0625.34044 · doi:10.1088/0305-4470/20/2/014 [5] DOI: 10.1080/16073606.1985.9631915 · Zbl 0618.34009 · doi:10.1080/16073606.1985.9631915 [6] DOI: 10.1080/16073606.1989.9632170 · Zbl 0683.34004 · doi:10.1080/16073606.1989.9632170 [7] DOI: 10.1063/1.528096 · Zbl 0785.34031 · doi:10.1063/1.528096 [8] DOI: 10.1063/1.526766 · Zbl 0587.34004 · doi:10.1063/1.526766 [9] DOI: 10.1016/0167-2789(87)90122-9 · Zbl 0656.70014 · doi:10.1016/0167-2789(87)90122-9 [10] DOI: 10.1017/S0022377800002737 · doi:10.1017/S0022377800002737 [11] DOI: 10.1063/1.526750 · Zbl 0552.70016 · doi:10.1063/1.526750 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.