The Dripping Faucet as a Model Chaotic System

Comments on "The dripping faucet as a model chaotic system," Shaw, Robert, Publisher - Science Frontier Express Series (Aerial Press, Inc.) 1984.

This is a technical story of how a dripping faucet can make you lose your sleep :-)

The major question to be addressed is: How do we construct a model from a stream of experimental data which we have not seen before? How do we use the model to make predictions? what are the limits of our predictive ability?

Chaotic dynamics - short-term predictability but long-term unpredictability. The system state at one instant of time is causally disconnected with its state far enough into the future.

The author wants to find answers to:
1. the amount of information a system is capable of storing, or transmitting from one instant of time to the next,
2. the rate of loss of this information.

If we consider any system to be a black box which is producing a stream of numbers - the above questions along with the "predictability" can be posed.

One method is to construct "return map" or Poincaire' cross-section by using a embedding dimension on time-delayed phase space reconstruction.

Most important attribute of the "characteristic of a system" is that quantities - "information stored in a system" - must be properties of the system, and not type of measurement. This implies an invariance under coordinate transformations - a property which appropriately defined measures of information possess.

The dripping faucet has different patterns of behavior - periodic or chaotic - depending on initial conditions, indicating multiple "basics of attraction", changes in behavior are sudden and hysteretic.

A behavior which examplifies a mixture of periodic and chaotic aspects can be called as noisy periodicity (May) or semiperdicity (Lorenz).

then there is a possibility of "mixed dimensionality" - in some regions of the state space - the state of the system can be optimally specified by the value of a single coordinate - but in othe rregions, more coordinates are required.

Interesting conclusions (page 18-19) made by the author are:
1. a model of only a few dimensions can sometimes adequately describe the chaotic behavior of a continuum system, the high dimensionality of the system is not required.
2. there exist fundamental geometrical structures which reappear in many different nonlinear systems.

Pg 30 -
Poincare realized that even in celestial mechanics determinism did not imply unlimited predictability. For purely deterministic dynamical systems, predictability may be extremely limited.
The degree of unpredictability --> entropy of the given system.

pg 32 ->
Kolmogorov, Sinai and others were able to show that, if one considers the set of all possible pratitions, and selects the one which gives the largest numerical value for the entropy, the number one obtains is a topological invariant of the system, independent of the coordinate system used to describe the original continuous variables v.

34:
the entropy describes the rate at which information flows from the microscopic variables up to the macropscopic. But in presence of noise, partition definition of entropy does not help,as the partition becomes finer than length scale by the noise.
In mechanics the transmission of information requires the possibility of change.

The channel capacity is defined as maximum rate one can find by varying the statistics of the input message over all possible input ensembles and using the optimum.
For dynamical systems, a particular input ensemble is selected by the properties of the system itself. Thus, information (Stored in the system) is quantifies as the average increase in out ability to predict the future of a system when we learn its past. Thus it is the difference in the expected randomness of a system with and without knowledge of its past. [pg 40]

[48-49]
our information about a system cannot increase in the absence of observation.
sharper distributions representing greater stored information will spread faster than broad distributions.
The maximum entropy or loss of predictability will thus occur when the stored information is at the maximum. under this definition both the stored information and entropy of a purely stochastic map will be zero.

pg 55 the entropy in the limit of "pure determinism"
In deterministic case (for zero noise), the information storage capacity of the map diverges. But, it becomes finite with the addition of noise, and information stored by higher iterates of the map has the concave behavior.

one task of an experimenter studying a new system is to characterize the determinism of the system, which can be described as the information stored through time in the dynamical variables. if the system has a positive entropy, some or all of this causal connection will be lost in the passage of time.

pg69
in a real chaotic system with noise present, initial data has predictive value for only a limited time, events which are too far apart are causally disconnected. correlations extend over only a finite length of the string of symbols, old experience ceases to have predictive value.

[pg 93]
from a classical perspective, it is clear that an experimenter's claim that he is "in the noise", and that there is no point in attempting to increase resolution, is an assumption. if the observation rate is less than the entropy of the system at some resolution level, the system will appear stochastic, even if it were "completely deterministic".

pg 103
The dimension of an object should describe how its volume scales with increasing linear size.

pg 109:
the usefulness of a "dimension" number as a scaling exponent depends on the number of orders of magnitude between the smallest and largest length scales. Robert Shaw comments that "the more low-dimensional structure there is in the reconstructed attractor, the less well-defined is a single "dimension" number. Thus, there is little reason at this point to prefer one dimension algorithm over another as "more fundamental".

In short:
Comments on the book -
Book has been vividly written with less of mathematical probing (or the probe is kept to minimal and necessary, whenever required).

Technically - attacks fundamental problem of what information and flow of information is related to "determinism" factor of a chaotic system.