The basic concepts of the mathematical theory of chaos are presented through a brief analysis of some interesting dynamical systems in one-, two- and three-dimensional space. We start with a discussion of interval maps and observe that when such maps are monotonic, their iterates behave in an orderly fashion. Then, by way of contrast, we study a well-known quadratic1 map iterates clearly manifest the archetypal characteristics of chaos, such as period-doubling bifurcations and the existence of a strange attractor. As a means of indicating that mappings in two dimensions yield a richer variety of chaotic regimes than do interval maps, we next discuss the horseshoe and solenoidal mappings of the two-disk. Dizzying forms of chaos emerge from these mappings, but there is an irony-the chaotic behavior can be characterized in an orderly way. We conclude with a cursory examination of the Lorenz differential equation in three-space: a primary source of the recent interest in chaos theory.
All Science Journal Classification (ASJC) codes
- Modeling and Simulation
- Computational Theory and Mathematics
- Computational Mathematics