Nonchaotic and Chaotic Behavior in Three-Dimensional Quadratic Systems: Five-One Conservative Cases

In this paper we study the nonchaotic and chaotic behavior of all 3D conservative quadratic ODE systems with five terms on the right-hand side and one nonlinear term (5-1 systems). We prove a theorem which provides sufficient conditions for solutions in 3D autonomous systems being nonchaotic. We show that all but five of these systems:(3.8a,b), (3.11b), (3.34)(A = ∓1), (4.1b),and (4.7a,b) are nonchaotic. Numerical simulations show that only one of the five systems, (4.1b), really appears to be chaotic. If proved to be true, it will be the simplest ODE system having chaos.