Chaos and Nonlinearity

As we have just discussed, Earth is representative of a nonlinear system. For example, the convection processes that occur in the Earth's core and mantle, the atmosphere and the ocean represent a nonlinear system. These processes were simulated in the famous Benard experiment which demonstrated that the system formed by a simple layer of fluid heated from below shows interesting dynamic behavior.

Convection is a deterministic system in the sense that there is a unique, predictable termperature gradient (the critical gradient) at which the liquid begins to covect, and also stochastic in the sense that there is no possible way to predict which cells will rotate clockwise and which counterclockwise. In addition, the behavior of the system changes radically from steady conduction to fully developed, vigourous convection at the moment the temperature reaches the critical value. These are characteristics of a nonlinear system, but also of a cooperation between chance and determinism, a duality that appears in nature in other contexts, such as biological evolution, where random mutations and deterministic natural selection combine to produce diversity and change.

Click here for an animation of convection

Nonlinearity can give rise to unexpected structures and events in the form of abrupt transitions, a multiplicity of states (e.g., the two possible rotation directions of the convection cells), pattern formation, or even a totally irregurlar (random-like) evolution in space and time, known as chaos. A chaotic dynamic system is one in which small changes can induce large and unpredictable effects, a condition normally defined as "strong dependence on initial conditions." All chaotic systems must be nonlinear, but nonlinear systems are not necessarily chaotic.

Chaos exists in the dancing flames of a camp fire, in the capricious forms of the clouds, in the swirls of a turbulent white water river, in the complex convection currents of the mantle and the core, in the size and distribution of earthquakes, in the craziness of the weather. On the other hand, in a linear system there is no chaos, there is no multiplicity of states (i.e., only one thing can happen next), no pattern is formed that was not there originally, no abrupt transitions occur, and everything is perfectly predictable.

It is no exaggeration to say that one of the greatest challenges facing science today, and possible well into the next century, is the accurate and complete description of complex, nonlinear systems.