Chaos Theory

NONLINEAR DYNAMICAL SYSTEMS that have sensitive dependence on initial conditions may exhibit chaotic behavior. In other words, if initial conditions are available only with some finite precision, two solutions starting from undistinguishable initial conditions (i.e., whose difference is smaller than the precision) can exhibit completely different future evolutions after time. Thus, the system behavior is unpredictable. Sensitive dependence on initial conditions can occur even in deterministic systems whose solutions are not influenced by any stochastic effects. Chaos theory attempts to find an underlying order in such chaotic behavior

In the early 1900s, H. Poincaré noticed that simple nonlinear deterministic systems can behave in a chaotic fashion. While studying the three-body problem in celestial mechanics, he found that the evolution of three planets could be complex and sensitive to their relative initial positions. Other early pioneers in chaotic dynamics from a mathematical viewpoint include G. Birkhoff, M.T. Cartright, J.E. Littlewood, S. Smale, and A.N. Kolmogorov, for a wider range of physical systems. Experimentalists also observed chaotic behaviors in turbulent fluid motion and radio circuits.

Weather is, without a doubt, one of the best examples of chaotic behaviors in nature. Future weather can never be predicted correctly beyond a certain period of time. The first identification of chaotic behavior in atmospheric sciences was made accidentally by E. Torenz in 1960s. To study the problem of predicting atmospheric convection, he developed a simple deterministic model with 12 variables. Simplified models such as these are quite useful in shedding light on the corresponding complex system. He followed a numerical approach by solving the model on a digital computer. On one occasion, he repeated a numerical experiment using an initial condition that was rounded-off to the first three digits (e.g., 0.506) from the original six digits (e.g., 0.506127), expecting that the difference between the two solutions would remain very small. To his surprise, the new solution stayed close to the original one for about a month of the model simulation, and then suddenly transitioned to a completely different behavior. This discovery has come to be known as the “butterfly effect.”

By repeating similar experiments, Lorenz also noticed that transitions happened randomly, irrespective of the size of the perturbations in the initial conditions. To conduct further investigations, Lorenz developed a simpler model, with three variables, able to mimic the chaotic behavior of the 12-variable model, in terms of sensitivity to the dependence on [Page 187]initial conditions. By plotting the solutions, he discovered another astonishing aspect of chaotic behavior: all solutions were attracted to a densely nested curve with a shape of double spirals. Starting from an arbitrary initial condition, a solution eventually reached the double spirals and never left them. It looped around one spiral, occasionally made a transition to the other spiral, then looped around, returning to the first spiral, and so on. The curve clearly moderates the chaotic behaviors of this model and is called the strange attractor. Qualitative and quantitative description of chaotic behavior can be made by understanding the properties of the strange attractor.

NASA hurricane animation may help predict storms more accurately. Weather is the best example of a chaotic behavior.

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