We present a theory of constructive Poincaré maps. The basis of our theory is the concept of irreducible nonlinear maps closely associated to concepts from Lie groups. Irreducible nonlinear maps are, heuristically, nonlinear maps which cannot be made simpler without removing the nonlinearity. A single irreducible map cannot produce chaos or any complex nonlinear effect. It can be implemented in an electronic circuit, and there are only a finite number of families of irreducible maps in any n-dimensional space. The composition of two or more irreducible maps can produce chaos and most of the maps studied today that produce chaos are compositions of two or more irreducible maps. The composition of a finite number of irreducible maps is called a completely reducible map and a map which can be approximated pointwise by completely reducible maps is called a reducible map. Poincaré maps from sinusoidally forced oscillators are the most familiar examples of reducible maps. This theoretical framework provides an approach to the construction of "closed form" Poincaré maps having the properties of Poincaré maps of systems for which the Poincaré map cannot be obtained in closed form. In particular, we derive a three-dimensional ODE for which the Hénon map is the Poincaré map and show that there is no two-dimensional ODE which can be written down in closed form for which the Hénon map is the Poincaré map. We also show that the Chirikov (standard) map is a Poincaré map for a two-dimensional closed form ODE. As a result of our theory, these differential equations can be mapped into electronic circuits, thereby associating them with real world physical systems. In order to clarify our results with respect to the abstract mathematical concept of suspension, which says that every C1 invertible map is a Poincaré map, we introduce the concept of a constructable Poincaré map. Not every map is a constructable Poincaré map and this is an important distinction between dynamical synthesis and abstract nonlinear dynamics. We also show how to use any one-dimensional map to induce a two-dimensional Poincaré map which is a completely reducible map and hence for a very broad class of maps that includes the logistic map we derive closed form ODEs for which these one-dimensional maps are "embedded" in a Poincaré map. This provides an avenue for the study of one-dimensional maps, such as the logistic map, as two-dimensional Poincaré maps that arise from square-wave forced electronic circuits.
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