## Abstract

One model of randomness observed in physical systems is that low-dimensional deterministic chaotic attractors underly the observations. A phenomenological theory of chaotic dynamics requires an accounting of the information flow fromthe observed system to the observer, the amount of information available in observations, and just how this information affects predictions of the system's future behavior. In an effort to develop such a description, the information theory of highly discretized observations of random behavior is discussed. Metric entropy and topological entropy are well-defined invariant measures of such an attractor's level of chaos, and are computable using symbolic dynamics. Real physical systems that display low dimensional dynamics are, however, inevitably coupled to high-dimensional randomness, e.g. thermal noise. We investigate the effects of such fluctuations coupled to deterministic chaotic systems, in particular, the metric entropy's response to the fluctuations. It is found that the entropy increases with a power law in the noise level, and that the convergence of the entropy and the effect of fluctuations can be cast as a scaling theory. It is also argued that in addition to the metric entropy, there is a second scaling invariant quantity that characterizes a deterministic system with added fluctuations: I/sub 0/, the maximum
More>>

## Citation Formats

Crutchfield, J P, and Packard, N H.
Symbolic dynamics of noisy chaos.
Netherlands: N. p.,
1983.
Web.

Crutchfield, J P, & Packard, N H.
Symbolic dynamics of noisy chaos.
Netherlands.

Crutchfield, J P, and Packard, N H.
1983.
"Symbolic dynamics of noisy chaos."
Netherlands.

@misc{etde_5883627,

title = {Symbolic dynamics of noisy chaos}

author = {Crutchfield, J P, and Packard, N H}

abstractNote = {One model of randomness observed in physical systems is that low-dimensional deterministic chaotic attractors underly the observations. A phenomenological theory of chaotic dynamics requires an accounting of the information flow fromthe observed system to the observer, the amount of information available in observations, and just how this information affects predictions of the system's future behavior. In an effort to develop such a description, the information theory of highly discretized observations of random behavior is discussed. Metric entropy and topological entropy are well-defined invariant measures of such an attractor's level of chaos, and are computable using symbolic dynamics. Real physical systems that display low dimensional dynamics are, however, inevitably coupled to high-dimensional randomness, e.g. thermal noise. We investigate the effects of such fluctuations coupled to deterministic chaotic systems, in particular, the metric entropy's response to the fluctuations. It is found that the entropy increases with a power law in the noise level, and that the convergence of the entropy and the effect of fluctuations can be cast as a scaling theory. It is also argued that in addition to the metric entropy, there is a second scaling invariant quantity that characterizes a deterministic system with added fluctuations: I/sub 0/, the maximum average information obtainable about the initial condition that produces a particular sequence of measurements (or symbols). 46 references, 14 figures, 1 table.}

journal = {Physica D (Amsterdam); (Netherlands)}

volume = {7D:1/3}

journal type = {AC}

place = {Netherlands}

year = {1983}

month = {May}

}

title = {Symbolic dynamics of noisy chaos}

author = {Crutchfield, J P, and Packard, N H}

abstractNote = {One model of randomness observed in physical systems is that low-dimensional deterministic chaotic attractors underly the observations. A phenomenological theory of chaotic dynamics requires an accounting of the information flow fromthe observed system to the observer, the amount of information available in observations, and just how this information affects predictions of the system's future behavior. In an effort to develop such a description, the information theory of highly discretized observations of random behavior is discussed. Metric entropy and topological entropy are well-defined invariant measures of such an attractor's level of chaos, and are computable using symbolic dynamics. Real physical systems that display low dimensional dynamics are, however, inevitably coupled to high-dimensional randomness, e.g. thermal noise. We investigate the effects of such fluctuations coupled to deterministic chaotic systems, in particular, the metric entropy's response to the fluctuations. It is found that the entropy increases with a power law in the noise level, and that the convergence of the entropy and the effect of fluctuations can be cast as a scaling theory. It is also argued that in addition to the metric entropy, there is a second scaling invariant quantity that characterizes a deterministic system with added fluctuations: I/sub 0/, the maximum average information obtainable about the initial condition that produces a particular sequence of measurements (or symbols). 46 references, 14 figures, 1 table.}

journal = {Physica D (Amsterdam); (Netherlands)}

volume = {7D:1/3}

journal type = {AC}

place = {Netherlands}

year = {1983}

month = {May}

}