Solutions of Sakaki-Kakei equations of type 1, 2, 7 and 12

Abstract

We consider solutions of Sakaki-Kakei equations of type 1, 2, 7 and 12, which are irreversible two dimensional systems. We first obtain their conserved quantities, and reduce them to one dimensional nonautonomous systems. We next show that the equations of type 2 and 7 are transformed to arithmetic-harmonic mean equation, and obtain their general solutions. We finally show that the equations of type 1 and 12 are related to the solvable chaotic system proposed by Umeno. We also show that their iteration maps have self semiconjugacy, and obtain their particular solutions which are expressed in terms of lemniscate elliptic function.