Integrable discrete dynamics and fibonacci sequences

Mathematical entities associated with the golden mean, Fibonacci sequences and related concepts exhibit themselves abundantly in a variety of natural processes - perhaps most notably in plant phyllotaxis. The work in this paper represents an effort to obtain a simple, integrable, discrete dynamical system model for phyllotaxis that naturally establishes the preeminence of the golden mean and is inclusive enough to subsume many of the numerous mathematical aspects of plant growth phenomena. A deformation energy function, inspired by certain geometric properties observed in plant morphology, is introduced on the set of smooth, structurally stable diffeomorphisms of the n-torus. It is proved that the minimizers of the deformation energy comprise a denumerable hierarchy of diffeomorphisms characterized by their identification as harmonic mappings. The discrete dynamical systems generated by these diffeomorphisms are all integrable, and they include the systems corresponding to the Fibonacci sequence and its variants. It is demonstrated how this dynamical systems context for plant phyllotaxis provides an effective mechanism for deriving the well-known connections of this biological process with such mathematical concepts as the golden mean, the noble numbers, the Farey tree, chaos theory and symbolic dynamics.