Defining Jacobian elliptic functions via nonpolynomial differential equations

Abstract

Although Jacobi elliptic functions have been known for almost two centuries, they are still the subject of intensive investigation. In this paper, contrary to the usual definition, we prove that the Jacobi elliptic functions can be defined by using nonconservative equations with limit cycles through existence theorems involving first integrals. This allows extending their validity domains, that is, their range of applications.

Nonautonomous Lienard systems equivalent to first-order differential equations

“…In this section, a brief review of the theory of second-order differential equations based on the first integral, recently introduced in the literature by the present authors, is presented [6,[15][16][17][18][19].…”