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MATLAB PROGRAMMING FOR VISUALIZING NUMERICAL ITERATIVE DYNAMICS

Abstract Presented here are the fundamental principles of discrete dynamical system and chaos from the point of view of numerical iterative algorithms. Also included is the visualization of some of such dynamical systems using Matlab programs. Such a visualization has a profound impact on our conceptual clarity and knowledge of the real world scientific and engineering problems.

1. Introduction

The observation that determinism may not imply predictable or regular (or periodic) behaviour influenced many areas of science and engineering. This observation led to the discovery of chaos that has had a significant impact on our conceptual clarity and knowledge of the real-world problems. The purpose of this article is to introduce the fundamental principles of discrete dynamical systems and chaos from the point of view of numerical algorithms and visualizing such dynamical systems using Matlab. This subject throws more light on the chaotic physical phenomena arising in the biochemical reactions, lasers, fluid flow, hurricane, tornedo, and earthquake. The study of chaos or, more generally nonlinear dynamical systems, became a fascinating area since early 1960’s (Lorenz 1963; Baker and Gollub 1996; Flake 2001). The dictionary meaning of the word “chaos” is complete disorder or confusion. Chaos in science and engineering refers to an apparent lack of order in a system that nevertheless obeys certain laws and rules. This understanding of chaos is the same as that of dynamical instability. Deterministic system can produce results which are chaotic and appear to be random. But these are not technically random because the events can be modeled by a nonlinear procedure/formula. The pseudo-random number generator in a computer is an example of such a system. A system which is stable, linear or non-chaotic under certain conditions may degenerate into randomness or unpredictability (may be partial) under other conditions. A pattern may still be discovered in this system on a different level. In chaos versus randomness, the common characteristic is unpredictability (at least partial). Observe that absolute randomness does not exist either in material universe/nature or in artificial environment (e.g., in computer) since any outcome has to follow the laws of nature and certain procedures. Chaos is the irregular and unpredictable time evolution of a nonlinear dynamical system; the system does not repeat its past behaviour even approximately. It occurs in rotating fluids, mechanical oscillators, and some chemical reactions. In spite of its irregularity, chaotic behaviour obeys a deterministic set of equations such as those based on Newton’s second law of motion. For example, if we start a a nonchaotic system twice with slightly different initial conditions, the uncertainty (due to, say, measurement errors) leads only to an error in prediction that grows linearly with time and the state of the system is known after a short time. However, in a chaotic dynamical system, the error grows exponentially with time so that the state of the system is not known after a short time. Discrete dynamical systems are essentially iterated function systems and hence computers are best suited for this monotonous simple task of simulating such systems and producing beautiful images and visualizing them. Specifically, we will be concerned with the relationship between iterative numerical methods and dynamical systems. We will

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Sen, S., & Shaykhian, G. A. (2008, June), Matlab Programming For Visualizing Numerical Iterative Dynamics Paper presented at 2008 Annual Conference & Exposition, Pittsburgh, Pennsylvania. 10.18260/1-2--3661