Integrability and limit cycles of the Moon–Rand system

•The Moon-Rand differential system (MRDS) depends on 3 parameters•The MRDS is a quadratic polynomial differential system (QPDS) in R3R3.•We perturb the Moon–Rand system inside the class of all quadratic polynomial differential system in R3R3.•We use averaging theory for studying the Hopf bifurcation•Four periodic solutions can appear in the Hopf bifurcation.

We study the Darboux integrability of the Moon–Rand polynomial differential system. Moreover we study the limit cycles of the perturbed Moon–Rand system bifurcating from the equilibrium point located at the origin, when it is perturbed inside the class of all quadratic polynomial differential systems in R3R3, and we prove that at first order in the perturbation parameter ε   the perturbed system can exhibit one limit cycle, and that at second order it can exhibit four limit cycles bifurcating from the origin. We provide explicit expressions of these limit cycles up to order O(ε2)O(ε2).