Diffeomorphical equivalence vs topological equivalence among Sprott systems

In 1994, Sprott [Phys. Rev. E 50, 647-650 (1994)] proposed a set of 19 different simple dynamical systems producing chaotic attractors. Among them, 14 systems have a single nonlinear term. To the best of our knowledge, their diffeomorphical equivalence and the topological equivalence of their chaotic attractors were never systematically investigated. This is the aim of this paper. We here propose to check their diffeomorphical equivalence through the jerk functions, which are obtained when the system is rewritten in terms of one of the variables and its first two derivatives (two systems are thus diffeomorphically equivalent when they have exactly the same jerk function, that is, the same functional form and the same coefficients). The chaotic attractors produced by these systems-for parameter values close to the ones initially proposed by Sprott-are characterized by a branched manifold. Systems B and C produce chaotic attractors, which are observed in the Lorenz system and are also briefly discussed. Those systems are classified according to their diffeomorphical and topological equivalence.