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Chaotic and Nonchaotic Behavior in Three-Dimensional Quadratic Systems: 5-1 dissipative Cases

We show analytically that almost all three-dimensional dissipative quadratic systems of ordinary differential equations with a total of five terms on the right-hand side and one nonlinear term (namely 5-1 cases) are not chaotic except twenty one of them. Indeed we find nine systems that exhibit chaos, which were discovered by Sprott and Malasoma earlier. They are the simplest dissipative chaotic systems found so far. In this paper, we also extend Heidel–Zhang's theorem which provides sufficient conditions for solutions in the three-dimensional autonomous systems with polynomials and rational expressions on the right-hand side being nonchaotic. We then investigate the twenty one systems analytically and numerically. We show the portraits of some typical chaotic and nonchaotic solutions in phase space. For two of the systems that exhibit chaos we found stable period 1, 2, 4, 8 and 12 orbits numerically.