Integrability analysis of the Shimizu-Morioka system

The aim of this paper is to give some new insights into the Shimizu-Morioka system (x) over dot = y, (y) over dot = x - lambda y - xz, (z) over dot = -alpha z + x(2), from the integrability point of view. Firstly, we propose a linear scaling in time and coordinates which converts the Shimizu-Morioka system into a special case of the Rucklidge system when alpha not equal 0 and discuss the relationship between Shimizu-Morioka system and Rucklidge system. Based on this observation, Darboux integrability of the Shimizu-Morioka system with alpha not equal 0 is trivially derived from the corresponding results on the Rucklidge system. When alpha not equal 0, we investigate Darboux integrability of the Shimizu-Morioka system by the Grobner basis in algebraic geometry. Secondly, we use the stability of the singular points and periodic orbits to study the nonexistence of global C-1 first integrals of the Shimizu-Morioka system. Finally, in the case alpha not equal 0, we prove it is not rationally integrable for almost all parameter values by an extended Morales-Ramis theory, and in the case alpha not equal 0, we show that it is not algebraically integrable by quasi-homogeneous decompositions and Kowalevski exponents. Our results are in accord with the fact that this system admits chaotic behaviors for a large range of its parameters. (C) 2019 Elsevier B.V. All rights reserved.