21,645 research outputs found
Connecting period-doubling cascades to chaos
The appearance of infinitely-many period-doubling cascades is one of the most
prominent features observed in the study of maps depending on a parameter. They
are associated with chaotic behavior, since bifurcation diagrams of a map with
a parameter often reveal a complicated intermingling of period-doubling
cascades and chaos. Period doubling can be studied at three levels of
complexity. The first is an individual period-doubling bifurcation. The second
is an infinite collection of period doublings that are connected together by
periodic orbits in a pattern called a cascade. It was first described by
Myrberg and later in more detail by Feigenbaum. The third involves infinitely
many cascades and a parameter value of the map at which there is chaos.
We show that often virtually all (i.e., all but finitely many) ``regular''
periodic orbits at are each connected to exactly one cascade by a path
of regular periodic orbits; and virtually all cascades are either paired --
connected to exactly one other cascade, or solitary -- connected to exactly one
regular periodic orbit at . The solitary cascades are robust to large
perturbations. Hence the investigation of infinitely many cascades is
essentially reduced to studying the regular periodic orbits of . Examples discussed include the forced-damped pendulum and the
double-well Duffing equation.Comment: 29 pages, 13 figure
Towards Evaluating the Quality of a Spreadsheet: The Case of the Analytical Spreadsheet Model
We consider the challenge of creating guidelines to evaluate the quality of a
spreadsheet model. We suggest four principles. First, state the domain-the
spreadsheets to which the guidelines apply. Second, distinguish between the
process by which a spreadsheet is constructed from the resulting spreadsheet
artifact. Third, guidelines should be written in terms of the artifact,
independent of the process. Fourth, the meaning of "quality" must be defined.
We illustrate these principles with an example. We define the domain of
"analytical spreadsheet models", which are used in business, finance,
engineering, and science. We propose for discussion a framework and terminology
for evaluating the quality of analytical spreadsheet models. This framework
categorizes and generalizes the findings of previous work on the more narrow
domain of financial spreadsheet models. We suggest that the ultimate goal is a
set of guidelines for an evaluator, and a checklist for a developer.Comment: Proc. European Spreadsheet Risks Int. Grp. (EuSpRIG) 2011 ISBN
978-0-9566256-9-
Equicontinuous Families of Markov Operators in View of Asymptotic Stability
Relation between equicontinuity, the so called e property and stability of
Markov operators is studied. In particular, it is shown that any asymptotically
stable
Markov operator with an invariant measure such that the interior of its
support is nonempty satisfies the e property
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