The global dynamics of a 3-dimensional differential system in ℝ3 via a Darboux invariant

Abstract

The differential system ẋ = ax − yz, ẏ = −by + xz, ż = −cz + x2, where a, b and c are positive real parameters, has been studied numerically due to the big variety of strange attractors that it can exhibit. This system has a Darboux invariant when c = 2b. Using this invariant and the Poincaré compactification technique we describe analytically its global dynamics.