One-dimensional nonlinear systems, although simple in form, are applicable in a surprisingly wide variety of engineering contexts. | Steven H. Isabelle et. Al. Nonlinear Maps. 2000 CRC Press LLC. http . Nonlinear Maps Steven H. Isabelle Massachusetts Institute of Technology Gregory W. Wornell Massachusetts Institute of Technology Introduction Eventually Expanding Maps and Markov Maps Eventually Expanding Maps Signals From Eventually Expanding Maps Estimating Chaotic Signals in Noise Probabilistic Properties of Chaotic Maps Statistics of Markov Maps Power Spectra of Markov Maps Modeling Eventually Expanding Maps with Markov Maps References Introduction One-dimensional nonlinear systems although simple in form are applicable in a surprisingly wide variety of engineering contexts. As models for engineering systems their richly complex behavior has provided insight into the operation of for example analog-to-digital converters 1 nonlinear oscillators 2 and power converters 3 . As realizable systems they have been proposed as random number generators 4 and as signal generators for communication systems 5 6 . As analytic tools they have served as mirrors for the behavior of more complex higher dimensional systems 7 8 9 . Although one-dimensional nonlinear systems are in general hard to analyze certain useful classes of them are relatively well understood. These systems are described by the recursion x n f x n - 1 j n g x n initialized by a scalar initial condition x 0 where f and g - are real-valued functions that describe the evolution of a nonlinear system and the observation of its state respectively. The dependence of the sequence x n on its initial condition is emphasized by writing x n fn x 0 where f n represents the n-fold composition of f with itself. Without further restrictions of the form of f and g this class of systems is too large to easily explore. However systems and signals corresponding to certain well-behaved maps f and observation functions g - can be rigorously analyzed. Maps of this type often generate .
