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Iterative decoding algorithms may be viewed as high-dimensional nonlinear dynamical systems, depending on a large number of parameters. In this work, we introduce a simplified description of several iterative decoding algorithms in terms of the a posteriori average entropy, and study them as a function of a single parameter that closely approximates the signal-to-noise ratio (SNR). Using this approach, we show that virtually all the iterative decoding schemes in use today exhibit similar qualitative dynamics. In particular, a whole range of phenomena known to occur in nonlinear systems, such as existence of multiple fixed points, oscillatory behavior, bifurcations, chaos, and transient chaos are found in iterative decoding algorithms. As an application, we develop an adaptive technique to control transient chaos in the turbo-decoding algorithm, leading to a substantial improvement in performance. We also propose a new stopping criterion for turbo codes that achieves the same performance with considerably fewer iterations.