Modelling natural phenomena via Dynamical System Theory has become commonplace amongst mathematicians, physicists, engineers, and the like. As such, research in this field is currently underway, with a notable focus on chaos. The Competitive Modes Conjecture is a relatively new approach in the field of chaotic Dynamical Systems, aiming to understand why a strange attractor is chaotic or not. Up till now, the Conjecture has only been used to study multipolynomial systems because of their simplicity. As such, the study of non-multipolynomial systems is sparse, filled with ambiguity, and lacks mathematical structure. This paper strives to rectify this dilemma, providing the mathematical background needed to rigorously apply a large set of non-multipolynomial systems to the Competitive Modes Conjecture. Examples of this new theory include application of Lorenz System, the Chua System, and the Wimol-Banlue System. As far as the authors are aware, any previous application of the latter two systems to the Conjecture has not been attempted. Therefore, this paper presents the first applications of a whole new set of dynamical systems to the Conjecture.