Book ChapterDOI
Period Three Implies Chaos
TLDR
In this article, a generalized logistic equation was used to model the distribution of points of impact on a spinning bit for oil well drilling, as mentioned if this distribution is helpful in predicting uneven wear of the bit.Abstract:
The way phenomena or processes evolve or change in time is often described by differential equations or difference equations. One of the simplest mathematical situations occurs when the phenomenon can be described by a single number as, for example, when the number of children susceptible to some disease at the beginning of a school year can be estimated purely as a function of the number for the previous year. That is, when the number x n+1, at the beginning of the n + 1st year (or time period) can be written
$${x_{n + 1}} = F({x_n}),$$
(1.1)
where F maps an interval J into itself. Of course such a model for the year by year progress of the disease would be very simplistic and would contain only a shadow of the more complicated phenomena. For other phenomena this model might be more accurate. This equation has been used successfully to model the distribution of points of impact on a spinning bit for oil well drilling, as mentioned if [8, 11] knowing this distribution is helpful in predicting uneven wear of the bit. For another example, if a population of insects has discrete generations, the size of the n + 1st generation will be a function of the nth. A reasonable model would then be a generalized logistic equation
$${x_{n + 1}} = r{x_n}[1 - {x_n}/K].$$
(1.2)read more
Citations
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Journal ArticleDOI
Simple mathematical models with very complicated dynamics
Robert M. May,Robert M. May +1 more
TL;DR: This is an interpretive review of first-order difference equations, which can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations.
Journal ArticleDOI
Ergodic theory of chaos and strange attractors
Jean-Pierre Eckmann,David Ruelle +1 more
TL;DR: A review of the main mathematical ideas and their concrete implementation in analyzing experiments can be found in this paper, where the main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions).
Journal ArticleDOI
Oscillation and Chaos in Physiological Control Systems
Michael C. Mackey,Leon Glass +1 more
TL;DR: First-order nonlinear differential-delay equations describing physiological control systems displaying a broad diversity of dynamical behavior including limit cycle oscillations, with a variety of wave forms, and apparently aperiodic or "chaotic" solutions are studied.
Journal ArticleDOI
An equation for continuous chaos
TL;DR: A prototype equation to the Lorenz model of turbulence contains just one (second-order) nonlinearity in one variable as mentioned in this paper, which allows for a "folded" Poincare map (horseshoe map).
Book
Nonlinear Systems: Analysis, Stability, and Control
TL;DR: In this article, the authors compare Linear vs. Nonlinear Control of Differential Geometry with Linearization by State Feedback (LSF) by using Linearization and Geometric Non-linear Control (GNC).
References
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Deterministic nonperiodic flow
TL;DR: In this paper, it was shown that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states, and systems with bounded solutions are shown to possess bounded numerical solutions.
Book ChapterDOI
Differentiable dynamical systems
TL;DR: A survey article on the area of global analysis defined by differentiable dynamical systems or equivalently the action (differentiable) of a Lie group G on a manifold M is presented in this paper.
Book ChapterDOI
On the existence of invariant measures for piecewise monotonic transformations
Andrzej Lasota,James A. Yorke +1 more
TL;DR: In this article, a class of piecewise continuous, piecewise C transforma-tions on the interval [0, 1] has been shown to have absolutely continuous invariant measures.
Journal ArticleDOI
On finite limit sets for transformations on the unit interval
TL;DR: An infinite sequence of finite or denumerable limit sets is found for a class of many-to-one transformations of the unit interval into itself and the structure and order of occurrence is universal for the class.