Statistical properties of deterministic Bernoulli flows
This thesis presents several new theorems about the stability and the statistical properties of deterministic chaotic flows. Many concrete systems known to exhibit deterministic chaos have so far been shown to be of a class known as Bernoulli Flows. This class of flows is characterized by the Finitely Determined property, which can be checked in specific cases. The first theorem says that these flows can be modeled arbitrarily well for all time by continuous-time finite state Markov processes. In other words it is theoretically possible to model the flow arbitrarily well by a computer equipped with a roulette wheel. There follows a stability result, which says that one can distort the measurements made on the processes without affecting the approximation. These results are than applied to the problem of distinguishing deterministic chaos from stochastic processes in the analysis of time series. The second part of the thesis deals with a specific set of examples. Although it has been possible to analyze specific systems to determine whether they lie in the class of Bernoulli systems, the standard techniques rely on the construction of expanding and contracting fibers in the phase space of the system. These fibers are then used to coordinatize the phase space and to prove the existence of a hyperbolic structure. Unfortunately such methods may fail in the general case, where smoothness conditions and a small singular set cannot be assumed. For example, suppose the standard billiard flow on a square table with a perfectly round obstacle, which is known to be Bernoulli, is replaced by a similar flow on a table with a bumpy fractal-like obstacle: a model perhaps closer to nature. It is shown that these fibers no longer exist and hence cannot be used in the standard manner to prove Bernoulliness or ergodicity. But, one can use the fact that the class of Bernoulli flows is closed in the d-bar metric to show that this billard flow with a bumpy obstacle is in fact Bernoulli.
- Research Organization:
- Stanford Univ., CA (United States)
- OSTI ID:
- 121361
- Resource Relation:
- Other Information: TH: Thesis (Ph.D.); PBD: 1992
- Country of Publication:
- United States
- Language:
- English
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