Polygons and Polynomials as Synoptic Tools for System Guidance

The purpose of this study is to construct a powerful topography for recording the evolution of any compound system composed of (n) subsystems. The essential idea is based on the fact that any regular (n) polygon (with n sides) acts as a vector operator which transforms the plane in a union of (n) concentric isosceles triangles wherein a complex phasor is inserted. Covariant and contravariant axes are introduced in each phasor to obtain equivalence with Bond-Graphs. In each triangle we can follow the behaviour of the correspondent subsystems by building their power balance.

The circular polynomials : Pⁿ(θ) = Π_k ^[a_k⁽θ – k2π/n)] play as an algebraic carrier for the transformation of the plane into a (n) star (with n branches). We represent any system working by a variable power triangle. The system coupling increases the stability, and accordingly we will choose the electrical machine as a particular anticipation modeling.