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Journal ArticleDOI

On finite limit sets for transformations on the unit interval

01 Jul 1973-Journal of Combinatorial Theory, Series A (Academic Press)-Vol. 15, Iss: 1, pp 25-44
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.
About: This article is published in Journal of Combinatorial Theory, Series A.The article was published on 1973-07-01 and is currently open access. It has received 521 citations till now. The article focuses on the topics: Limit superior and limit inferior & Limit (mathematics).
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Book
01 Jan 1982
TL;DR: This book is a blend of erudition, popularization, and exposition, and the illustrations include many superb examples of computer graphics that are works of art in their own right.
Abstract: "...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature

24,199 citations

Journal ArticleDOI
10 Jun 1976-Nature
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.
Abstract: First-order difference equations arise in many contexts in the biological, economic and social sciences. Such equations, even though simple and deterministic, can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations. There are consequently many fascinating problems, some concerned with delicate mathematical aspects of the fine structure of the trajectories, and some concerned with the practical implications and applications. This is an interpretive review of them.

6,118 citations

Book ChapterDOI
TL;DR: 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)

3,278 citations

Journal ArticleDOI
TL;DR: In this article, a large class of recursion relations xn+l = Af(xn) exhibiting infinite bifurcation is shown to possess a rich quantitative structure essentially independent of the recursion function.
Abstract: A large class of recursion relations xn+l = Af(xn) exhibiting infinite bifurcation is shown to possess a rich quantitative structure essentially independent of the recursion function. The functions considered all have a unique differentiable maximum 2. With f(2) - f(x) ~ Ix - 21" (for Ix - 21 sufficiently small), z > 1, the universal details depend only upon z. In particular, the local structure of high-order stability sets is shown to approach universality, rescaling in successive bifurcations, asymptotically by the ratio c~ (a = 2.5029078750957... for z = 2). This structure is determined by a universal function g*(x), where the 2"th iterate off, f("~, converges locally to ~-"g*(~nx) for large n. For ithe class of f's considered, there exists a A~ such that a 2"-point stable limit cycle including :7 exists; A~ - ~ ~ ~-" (~ = 4.669201609103... for z = 2). The numbers = and have been computationally determined for a range of z through their definitions, for a variety off's for each z. We present a recursive mechanism that explains these results by determining g* as the fixed-point (function) of a transformation on the class off's. At present our treatment is heuristic. In a sequel, an exact theory is formulated and specific problems of rigor isolated.

2,984 citations


Cites background from "On finite limit sets for transforma..."

  • ...} (12) For (11) and (12) to agree, one has a ~-1 ~ 1 - g*'(1) (13) where /L ~< 1 corresponds to A-shifting being mostly a displacement in the immediate environs of 2....

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  • ...) ~_ p[/z + ~'(1)]" (23) with p ~ 1, n-independent....

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  • ..., (1) a two-point cycle should now become stable....

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  • ...P~+I = bp~ (1) accurately describes the population growth so long as it remains dilute, with the solution p~ = pob ~....

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  • ...Combining (10) and (13), one has 3 ~ ]g*'(1)][1 - g*'(1)] 11~-1 (14) While (13) and (14) are crude, they are roughly correct for z >~ 2, but more important , indicate that g* ultimately determines everything, We now proceed to describe the situation more carefully, tacitly assuming convergence, and successively illustrating its details through consistency arguments....

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Journal ArticleDOI
TL;DR: In this paper, the role of functional equations to describe the exact local structure of highly bifurcated attractors is formally developed, and a hierarchy of universal functions, each descriptive of the same local structure but at levels of a cluster of 2>>\s points, is presented.
Abstract: The role of functional equations to describe the exact local structure of highly bifurcated attractors ofx n+1 =λf(x n ) independent of a specificf is formally developed. A hierarchy of universal functionsg r (x) exists, each descriptive of the same local structure but at levels of a cluster of 2 r points. The hierarchy obeysg r−1 (x)=−αg r(gr(x/α), withg=limr → ∞ gr existing and obeyingg(x) = −αg(g(x/α), an equation whose solution determines bothg andα. Forr asymptoticg r ∼ g − δ−r h * where δ > 1 andh are determined as the associated eigenvalue and eigenvector of the operator ℒ: $$\mathcal{L}\left[ \psi \right] = - \alpha \left[ {\psi \left( {g\left( {{x \mathord{\left/ {\vphantom {x \alpha }} \right. \kern- ulldelimiterspace} \alpha }} \right)} \right) + g'\left( {g\left( {{x \mathord{\left/ {\vphantom {x \alpha }} \right. \kern- ulldelimiterspace} \alpha }} \right)} \right)\psi \left( {{{ - x} \mathord{\left/ {\vphantom {{ - x} \alpha }} \right. \kern- ulldelimiterspace} \alpha }} \right)} \right]$$ We conjecture that ℒ possesses a unique eigenvalue in excess of 1, and show that this δ is the λ-convergence rate. The form (*) is then continued to allλ rather than just discreteλ r and bifurcation valuesΛ r and dynamics at suchλ is determined. These results hold for the high bifurcations of any fundamental cycle. We proceed to analyze the approach to the asymptotic regime and show, granted ℒ's spectral conjecture, the stability of theg r limit of highly iterated λf's, thus establishing our theory in a local sense. We show in the course of this that highly iterated λf's are conjugate tog r 's, thereby providing some elementary approximation schemes for obtainingλ r for a chosenf.

1,160 citations

References
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Book
01 Jan 1964
TL;DR: In this paper, the properties of non-linear transformations in Euclidean spaces are examined, and the invariant points, finite sets, and invariant subsets of the transformations and the means for obtaining them constructively are considered.
Abstract: S>The properties of certain non-linear transformations in Euclidean spaces--mainly in two or three dimensions--are examined. The transformations, generally of very special and simple algebraic form, are bounded and continuous, but in general many-to-one. The iteration of the transformations and the asymptotic and ergodic properties of the sequence of iterated points are of primary interest; but the invariant points, finite sets, and invariant subsets of the transformations and the means for obtaining them constructively are also considered. (D.C.W.)

93 citations


"On finite limit sets for transforma..." refers background in this paper

  • ...f(x) is continuous, single-valued, and piece-wise fY1) on [0, 11, and strictly positive on the open interval, withf(0) = f(1) = 0....

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Journal ArticleDOI

45 citations


Additional excerpts

  • ...< htrn) with associated patterns H(l)@(l)), HL2)(h(2)),....

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