The Competitive Modes Conjecture is a relatively new approach in the field of Dynamical Systems, aiming to understand chaos in strange attractors using Resonance Theory. 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 the Competitive Modes Conjecture to a certain set of non-multipolynomial systems. Afterwards, we provide an example of this new theory in the non-multipolynomial Wimol-Banlue Attractor, something that up to this point has not been possible as far as the authors know.

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In this paper, we study a multi-parameter Liénard polynomial system carrying out its global bifurcation analysis. To control the global bifurcations of limit cycle in this systems, it is necessary to know the properties and combine the effects of all its field rotation parameters. It can be done by means of the development of our bifurcational geometric method based on the application of a canonical system with field rotation parameters. Using this method, we present a solution of Hilbert's Sixteenth Problem on the maximum number of limit cycles and their distribution for the Liénard polynomial system. We also conduct some numerical experiments to illustrate the obtained results.@en