Rhombic Tilings and Primordia Fronts of Phyllotaxis

We introduce and study properties of phyllotactic and rhombic tilings on the cylin- der. These are discrete sets of points that generalize cylindrical lattices. Rhombic tilings appear as periodic orbits of a discrete dynamical system S that models plant pattern formation by stacking disks of equal radius on the cylinder. This system has the advantage of allowing several disks at the same level, and thus multi-jugate config- urations. We provide partial results toward proving that the attractor for S is entirely composed of rhombic tilings and is a strongly normally attracting branched manifold and conjecture that this attractor persists topologically in nearby systems. A key tool in understanding the geometry of tilings and the dynamics of S is the concept of pri- mordia front, which is a closed ring of tangent disks around the cylinder. We show how fronts determine the dynamics, including transitions of parastichy numbers, and might explain the Fibonacci number of petals often encountered in compositae.Comment: 33 pages, 10 picture

Commuting polynomials and polynomials with same Julia set

It has been known since Julia that polynomials commuting under composition have the same Julia set. More recently in the works of Baker and Eremenko, Fern\'andez, and Beardon, results were given on the converse question: When do two polynomials have the same Julia set? We give a complete answer to this question and show the exact relation between the two problems of polynomials with the same Julia set and commuting pairs

Rhombic Tilings and Primordia Fronts of Phyllotaxis

We introduce and study properties of phyllotactic and rhombic tilings on the cylin- der. These are discrete sets of points that generalize cylindrical lattices. Rhombic tilings appear as periodic orbits of a discrete dynamical system S that models plant pattern formation by stacking disks of equal radius on the cylinder. This system has the advantage of allowing several disks at the same level, and thus multi-jugate config- urations. We provide partial results toward proving that the attractor for S is entirely composed of rhombic tilings and is a strongly normally attracting branched manifold and conjecture that this attractor persists topologically in nearby systems. A key tool in understanding the geometry of tilings and the dynamics of S is the concept of pri- mordia front, which is a closed ring of tangent disks around the cylinder. We show how fronts determine the dynamics, including transitions of parastichy numbers, and might explain the Fibonacci number of petals often encountered in compositae

A Dynamical System for Plant Pattern Formation: A Rigorous Analysis

We present a rigorous mathematical analysis of a discrete dynamical system modeling plant pattern formation. In this model, based on the work of physicists Douady and Couder, fixed points are the spiral or helical lattices often occurring in plants. The frequent occurrence of the Fibonacci sequence in the number of visible spirals is explained by the stability of the fixed points in this system, as well as by the structure of their bifurcation diagram. We provide a detailed study of this diagram

The Geometric and Dynamic Essence of Phyllotaxis

We present a dynamic geometric model of phyllotaxis based on two postulates, primordia formation and meristem expansion. We find that Fibonacci, Lucas, bijugate and multijugate are all variations of the same unifying phenomenon and that the difference lies on small changes in the position of initial primordia. We explore the set of all initial positions and color-code its points depending on the phyllotactic type of the pattern that arises

Cantor goes Julia

this paper we will restrict to parameter values c 2 RM (`) along an external ray RM (`) of the Mandelbrot set of angle `, with ` strictly preperiodic. RM (`) lands on some c 0 2 M. The Julia set J c 0 is, in this case, locally connected. We refer to Douady and Hubbard [2] as a general reference for external rays of both the Mandelbrot set and Julia sets. In the dynamic plane, for c 2 RM (`) and t 2

The Accuracy of Symplectic Integrators

Introduction The symplectic integration of Hamiltonian dynamical systems is by now an established technique. Ruth [1] has developed explicit methods for separable systems; his approach was extended to fourth order by Candy and Rozmus [2]. Channel and Scovel [3] and Feng et al. [4] have derived methods based on the Taylor series expansion of the time map of a general Hamiltonian. Feng [5] contains a survey of the Chinese program and an important generalisation which includes many known methods as special cases. In [6] Feng provides a discussion of the philosophy and history of symplectic integration. For a summary of explicit symplectic integrators for separable Hamiltonians, see (2.2) and Table II. Standard integrators do not generally preserve the Poincar'e integral invariants of a Hamiltonian flow and cannot hope to capture the long-time dynamics of the system. Typically their numerical diffusion causes orbits to be attracted to elliptic orbits, or, coupled with forcing, cr