Angular homeostasis: III. The formalism of discrete orbits in ontogeny

Angular homeostasis. VIII. Pursuit of a slowly moving target in a plane: relevance to lateralization in cardiovascular ontogeny

We explore the pursuit in a plane of a target moving at constant slow speed in a straight line. Two models of the pursuit are given. In the continuous case, the pursuer is moving at constant speed and is subject to proportionate angular homeostasis with correction constant b. In the discrete version movement occurs at a constant speed in a sequence of straight line segments of constant length (called the step size, s) the end of the segments being called the vertices. The pattern considered is not the absolute position of the pursuer, but its distance and orientation relative to the target. Both the transients and the asymptotic orbit are addressed. A key quantity is r, the speed of the target expressed as a fraction of that of the pursuer. If the speed of the pursuer is defined as unity, r is also the ratio of the speeds. There exists a critical speed fraction, R(b,s), a function of b and s, that defines what the term slow designates. R(b,s), which has to be found numerically, has the following property. For r less than R(b,s), the asymptotic path is a simple closed curve. In the discrete case the vertices converge to a simple closed curve. The larger r, the more the path (or in the discrete analogue its set of vertices) departs from a circle, and the more eccentric the target is with respect to it. Interest centers on two issues. First we address the transient patterns of the path, notably whether or not the sense of any particular path (clockwise or counterclockwise) is the same throughout, or changes at some stage.(ABSTRACT TRUNCATED AT 250 WORDS)

Angular homeostasis VII: non-monotonic correction systems

We extend our model of angular homeostasis to correction functions that have a single maximum at a discrepant angle less than pi radians. We find that there are stable, and asymptotically stable, solutions that in general consist of self-intersecting curves. We investigate conditions for these curves to be periodic, and describe their symmetries. One typical pattern of such a closed curve involves a finite number of loops, each having a reflection axis of symmetry, with the complete curve having a cyclic rotation group. These bear a close resemblance to patterns found in lobulated biological structures (such as the petals of a flower or the primitive fetal hand). We further discuss implications for morphogenesis.

Angular homeostasis: IV. Polygonal orbits

Some properties are discussed of regular polygons that may result from angular homeostatic processes in stable orbit. To characterize these "homeostatic polygons" we need to discuss the winding number, the sidedness (integer, fractional and irrational), multiplicity, envelopes, and density. A regular (i.e., equilateral, equiangular) polygon may be closed in one revolution about its unique center, in multiple revolutions, or not at all. A homeostatic polygon can be generated only if all vertices are included in a single polygon, which occurs if and only if the number of vertices and the number of revolutions required to complete the polygon are relatively prime. For the homeostatic polygon to have a finite number of sides (without repeating itself) the angle subtended by any two successive vertices at the center must be a rational multiple of 2 pi. Biological implications of these properties are illustrated.