Chaotic Behaviour in Three Dimensional Quadratic Systems.

This thesis presents a part of the proof of that there is no chaos in three dimensional autonomous quadratic systems of four terms with one nonlinear term and without constant terms. The first step of the proof is identifying equivalent systems from all possible systems by the 3rd order permutation group and as a result 138 inequivalent patterns are found(refer to appendix A). And then for the solvable systems we show how to solve them. Some of the nonsolvable systems turn out to be 2nd order autonomous systems and they are resolved by analyzing the monotonicity of the solutions and/or using the Poincaré-Bendixon theorem. The main and difficult part of the proof is to prove that the nonsolvable 3rd order systems have no chaos. This thesis introduces a general theory of analyzing the behavior of the solutions of higher dimensional autonomous systems (n≥3) qualitatively in the phase space. Some sufficient conditions for systems to have no chaos are concluded by this theory. We also found out that there is a very close relation between coupled loops and the properties of the positive bounded limit sets and chaotic behaviour of a system. Because of limited time, only the seven dissipative nonsolvable 3rd order patterns are studied in this thesis. As an application of the theory, we proved in this thesis that three of them have no chaos. Two of the remaining four have coupled loops and thus the behavior of these two patterns is determined. The last two can be resolved by analyzing the monotonicity of the solutions. A system known to have chaos is also analyzed by this theory and this is a typical example of a chaotic system with coupled loops and with more than one positive bounded limit set