Integrability analysis of the Shimizu–Morioka system

•We show that a homothetic transformation converts the Shimizu–Morioka system into the Rucklidge system.•The results on the Shimizu–Morioka system can be obtained from the corresponding results on the Rucklidge system.•Integrability analysis of the Shimizu–Morioka system is performed. The aim of this paper is to give some new insights into the Shimizu–Morioka systemx˙=y,y˙=x−λy−xz,z˙=−αz+x2,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 α ≠ 0 and discuss the relationship between Shimizu–Morioka system and Rucklidge system. Based on this observation, Darboux integrability of the Shimizu–Morioka system with α ≠ 0 is trivially derived from the corresponding results on the Rucklidge system. When α=0, we investigate Darboux integrability of the Shimizu–Morioka system by the Gröbner basis in algebraic geometry. Secondly, we use the stability of the singular points and periodic orbits to study the nonexistence of global C1 first integrals of the Shimizu–Morioka system. Finally, in the case α ≠ 0, we prove it is not rationally integrable for almost all parameter values by an extended Morales-Ramis theory, and in the case α=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.