Time-reversal, tori families, and canards in the Sprott A and NE9 systems

Quadratic three-dimensional autonomous systems may display complex behavior. Studying the systems Sprott A and NE9, we find families of tori and periodic solutions both involving canards. Using time-reversal and symmetry, we are able to explain in these two systems both the analysis and origin of tori, periodic solutions, and the numerics of these objects. For system NE9, unbounded solutions exist that admit analytic description by singular perturbation theory of the flow near infinity, also we observe torus destruction and a new chaotic attractor (Kaplan-Yorke dimension 2.1544) produced by a period-doubling scenario. The subtle numerics of periodic solutions involving canards is explained in the final section.