Periodic and Elliptic Asymptotic Forms of Solutions to the Fifth Painlev'e Equation

The method of calculation of elliptic and periodic asymptotic forms of solutions to an ordinary differential equation of a quite general form is described in the first part of this work. It is described in the case when an independent variable is tending to infinity. Then we show how these asymptotic forms can be continued to the corresponding asymptotic expansions. Finally these methods are applied to the fifth Painlevé equation. We have obtained 2 famlies of elliptic asymptotic forms and 4 families of power-periodic expansions of solutions to the fifth Painlevé equation. All these families are 2-parameter.