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Properties of second-order ordinary differential equations invariant under time translation and self-similar transformation. (English) Zbl 0734.34029

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.

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
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