PROPERTIES OF 2ND-ORDER ORDINARY DIFFERENTIAL-EQUATIONS INVARIANT UNDER TIME TRANSLATION AND SELF-SIMILAR TRANSFORMATION

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.