Intriguing structures in iterative maps motivated by N-body problem

Abstract

Conventional approaches to modeling any system try to incorporate increasingly realistic features into the model, thereby making it more and more complex. An opposite approach seeks to build simpler and simpler conceptual models capable of capturing some observed features of a system. This trend began with Lorenz, who simplified models of the atmosphere to obtain the Lorenz model consisting of a system of only three equations. Despite the simplicity of these equations, this system displayed surprisingly rich properties, and has been used as a conceptual model in diverse disciplines. Poincare maps help study ordinary differential equations from a qualitative perspective. Several investigators like Henon and Feigenbaum followed this simplification approach. Instead of investigating Poincare maps of realistic systems, they, along with several others, investigated simple maps for their own sake. Despite lack of realism, this approach proved to be very fruitful. A map, as simple as the Logistic map, became an important conceptual modeling paradigm. It provided a tool for understanding bifurcation routes to chaos, which were verified experimentally through various experiments in diverse fields. Coupled map lattices (CML) help explore partial differential equations (PDE). Further simplification led to the introduction of Cellular Automata (CA). These fields continue to be explored with vigor and have given rise to a rich body of knowledge, conceptually useful over a wide spectrum of disciplines. In this paper, we follow the simplification approach for modeling the N-body problem. N-body simulations, say in Gravitation, give rise to filamentary structures. Such structures are observed in the actual observed Galactic distribution. The mechanism for creation of such structures is not well understood. We present a simple iterative dynamical model, motivated by the N-body problem, which, though unrealistic, produces such filamentary structures. This model also exhibits a variety of intriguing structures. Attempts to understand these structures may lead to useful insights similar to those provided by investigations in maps, CML, CA etc.