scholarly journals The doubling map with asymmetrical holes

2013 ◽  
Vol 35 (4) ◽  
pp. 1208-1228 ◽  
Author(s):  
PAUL GLENDINNING ◽  
NIKITA SIDOROV
Keyword(s):  
Hausdorff Dimension ◽  
Period Doubling ◽  
Second Order ◽  
Full Description ◽  
Natural Numbers ◽  
Routes To Chaos ◽  
Route To Chaos ◽  
Image Position ◽  
Doubling Map

AbstractLet $0\lt a\lt b\lt 1$ and let $T$ be the doubling map. Set $ \mathcal{J} (a, b): = \{ x\in [0, 1] : {T}^{n} x\not\in (a, b), n\geq 0\} $. In this paper we completely characterize the holes $(a, b)$ for which any of the following scenarios hold: (i) $ \mathcal{J} (a, b)$ contains a point $x\in (0, 1)$; (ii) $ \mathcal{J} (a, b)\cap [\delta , 1- \delta ] $ is infinite for any fixed $\delta \gt 0$; (iii) $ \mathcal{J} (a, b)$ is uncountable of zero Hausdorff dimension; (iv) $ \mathcal{J} (a, b)$ is of positive Hausdorff dimension. In particular, we show that (iv) is always the case if $$\begin{eqnarray*}b- a\lt \frac{1}{4} { \mathop{\prod }\nolimits}_{n= 1}^{\infty } (1- {2}^{- {2}^{n} } )\approx 0. 175\hspace{0.167em} 092\end{eqnarray*}$$ and that this bound is sharp. As a corollary, we give a full description of first- and second-order critical holes introduced by N. Sidorov [Supercritical holes for the doubling map. Preprint, see http://arxiv.org/abs/1204.1920] for the doubling map. Furthermore, we show that our model yields a continuum of ‘routes to chaos’ via arbitrary sequences of products of natural numbers, thus generalizing the standard route to chaos via period doubling.

The mean value theorem in second order arithmetic

2001 ◽  
Vol 66 (3) ◽  
pp. 1353-1358 ◽  
Author(s):  
Christopher S. Hardin ◽  
Daniel J. Velleman
Keyword(s):  
Mean Value ◽  
Second Order ◽  
Mean Value Theorem ◽  
Natural Numbers ◽  
Second Order Arithmetic ◽  
Elementary Analysis ◽  
Logical Connectives ◽  
The Mean ◽  
Order Arithmetic ◽  
Image Position

This paper is a contribution to the project of determining which set existence axioms are needed to prove various theorems of analysis. For more on this project and its history we refer the reader to [1] and [2].We work in a weak subsystem of second order arithmetic. The language of second order arithmetic includes the symbols 0, 1, =, <, +, ·, and ∈, together with number variables x, y, z, … (which are intended to stand for natural numbers), set variables X, Y, Z, … (which are intended to stand for sets of natural numbers), and the usual quantifiers (which can be applied to both kinds of variables) and logical connectives. We write ∀x < t φ and ∃x < t φ as abbreviations for ∀x(x < t → φ) and ∃x{x < t ∧ φ) respectively; these are called bounded quantifiers. A formula is said to be if it has no quantifiers applied to set variables, and all quantifiers applied to number variables are bounded. It is if it has the form ∃xθ and it is if it has the form ∀xθ, where in both cases θ is .The theory RCA0 has as axioms the usual Peano axioms, with the induction scheme restricted to formulas, and in addition the comprehension scheme, which consists of all formulas of the formwhere φ is , ψ is , and X does not occur free in φ(n). (“RCA” stands for “Recursive Comprehension Axiom.” The reason for the name is that the comprehension scheme is only strong enough to prove the existence of recursive sets.) It is known that this theory is strong enough to allow the development of many of the basic properties of the real numbers, but that certain theorems of elementary analysis are not provable in this theory. Most relevant for our purposes is the fact that it is impossible to prove in RCA0 that every continuous function on the closed interval [0, 1] attains maximum and minimum values (see [1]).Since the most common proof of the Mean Value Theorem makes use of this theorem, it might be thought that the Mean Value Theorem would also not be provable in RCA0. However, we show in this paper that the Mean Value Theorem can be proven in RCA0. All theorems stated in this paper are theorems of RCA0, and all of our reasoning will take place in RCA0.

Fractal bifurcation sets, renormalization strange sets and their universal invariants

1987 ◽  
Vol 413 (1844) ◽  
pp. 45-61 ◽  
Keyword(s):  
Hausdorff Dimension ◽  
Fractal Structure ◽  
Period Doubling ◽  
Invariant Sets ◽  
Transition Structure ◽  
Routes To Chaos

The transition structure of the most common routes to chaos are organized by fractal bifurcation sets. Examples include the quasiperiodic transitions to chaos and the period-doubling structure found in Arnol’d tongues. In this paper I discuss the universality of such fractal bifurcation sets and their relation to strange invariant sets of renormalization transformations. An important result is that fractal bifurcation sets from within the same universality chaos are lipeomorphic. This implies that they have the same fractal structure and, in particular, the same Hausdorff dimension and scaling spectra. Some other invariants are introduced.

Arithmetic and the theory of types

1984 ◽  
Vol 49 (2) ◽  
pp. 621-624 ◽  
Author(s):  
M. Boffa
Keyword(s):  
Peano Arithmetic ◽  
Positive Answer ◽  
Second Order ◽  
Countable Model ◽  
Natural Numbers ◽  
Conservative Extension ◽  
Logical Definition ◽  
Claude Bernard ◽  
Definition Of ◽  
Image Position

A hundred years ago, Frege proposed a logical definition of the natural numbers based on the following idea:He replaced this circular definition by the following one:He tried afterwards to found his theory over a notion of class satisfying a general comprehension principle:Russell quickly derived a contradiction from this principle (the famous Russell's paradox) but saved Frege's arithmetic with his theory of types based on the following comprehension principle:In 1979, talking at the Claude Bernard University in Lyon, I remarked that 3 types suffice to provide Frege's arithmetic, showing in fact that PA2 (second order Peano arithmetic) holds in TT3 + AI (theory of types 0, 1, 2 plus a suitable axiom of infinity). I asked whether TT3 + AI was a conservative extension of PA2. Pabion [3] gave a positive answer by a subtle use of the Fraenkel-Moskowski method. This result will be improved in the present paper, with a view to getting models of NF3 + AI in which Frege's arithmetic forms a model isomorphic to a given countable model of PA2.

A POSSIBLE NEW ROUTE TO CHAOS VIA SEMICONDUCTOR SUPERLATTICES

2007 ◽  
Vol 21 (23n24) ◽  
pp. 3967-3974
Author(s):  
X. R. WANG ◽  
Z. Z. SUN ◽  
ZHENYU ZHANG
Keyword(s):  
Limit Cycles ◽  
Logistic Map ◽  
Period Doubling ◽  
Current Understanding ◽  
Semiconductor Superlattices ◽  
Frequency Locking ◽  
Routes To Chaos ◽  
Route To Chaos ◽  
Geometric Picture ◽  
Torus Bifurcation

Our current understanding of routes to chaos is mainly based on torus bifurcation where new periods are generated, the period-doubling mechanism revealed in the logistic map, and intermittency where periodic and burst motion appear alternatively. We present a possible new route to chaos based on our geometric picture of the frequency-locking of limit-cycles in semiconductor superlattices. In the period-double route and/or its variations, the period increases exponentially with bifurcation order, whereas the period in the new route increases linearly with the order of bifurcations.

scholarly journals The -transformation with a hole at 0

2019 ◽  
Vol 40 (9) ◽  
pp. 2482-2514
Author(s):  
CHARLENE KALLE ◽  
DERONG KONG ◽  
NIELS LANGEVELD ◽  
WENXIA LI
Keyword(s):  
Hausdorff Dimension ◽  
Upper Bound ◽  
The Other ◽  
Bifurcation Set ◽  
Other Hand ◽  
Accumulation Points ◽  
Isolated Points ◽  
Image Position ◽  
Doubling Map ◽  
Null Set

For $\unicode[STIX]{x1D6FD}\in (1,2]$ the $\unicode[STIX]{x1D6FD}$-transformation $T_{\unicode[STIX]{x1D6FD}}:[0,1)\rightarrow [0,1)$ is defined by $T_{\unicode[STIX]{x1D6FD}}(x)=\unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$. For $t\in [0,1)$ let $K_{\unicode[STIX]{x1D6FD}}(t)$ be the survivor set of $T_{\unicode[STIX]{x1D6FD}}$ with hole $(0,t)$ given by $$\begin{eqnarray}K_{\unicode[STIX]{x1D6FD}}(t):=\{x\in [0,1):T_{\unicode[STIX]{x1D6FD}}^{n}(x)\not \in (0,t)\text{ for all }n\geq 0\}.\end{eqnarray}$$ In this paper we characterize the bifurcation set $E_{\unicode[STIX]{x1D6FD}}$ of all parameters $t\in [0,1)$ for which the set-valued function $t\mapsto K_{\unicode[STIX]{x1D6FD}}(t)$ is not locally constant. We show that $E_{\unicode[STIX]{x1D6FD}}$ is a Lebesgue null set of full Hausdorff dimension for all $\unicode[STIX]{x1D6FD}\in (1,2)$. We prove that for Lebesgue almost every $\unicode[STIX]{x1D6FD}\in (1,2)$ the bifurcation set $E_{\unicode[STIX]{x1D6FD}}$ contains infinitely many isolated points and infinitely many accumulation points arbitrarily close to zero. On the other hand, we show that the set of $\unicode[STIX]{x1D6FD}\in (1,2)$ for which $E_{\unicode[STIX]{x1D6FD}}$ contains no isolated points has zero Hausdorff dimension. These results contrast with the situation for $E_{2}$, the bifurcation set of the doubling map. Finally, we give for each $\unicode[STIX]{x1D6FD}\in (1,2)$ a lower and an upper bound for the value $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$ such that the Hausdorff dimension of $K_{\unicode[STIX]{x1D6FD}}(t)$ is positive if and only if $t<\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$. We show that $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}\leq 1-(1/\unicode[STIX]{x1D6FD})$ for all $\unicode[STIX]{x1D6FD}\in (1,2)$.

The period doubling route to chaos of a second order non-linear non-autonomous chaotic oscillator––part II

2004 ◽  
Vol 20 (4) ◽  
pp. 849-854 ◽  
Author(s):  
S.G. Stavrinides ◽  
K.G. Kyritsi ◽  
N.C. Deliolanis ◽  
A.N. Anagnostopoulos ◽  
A. Tamaševičious ◽  
...  
Keyword(s):  
Period Doubling ◽  
Second Order ◽  
Chaotic Oscillator ◽  
Non Linear ◽  
Route To Chaos

The period doubling route to chaos of a second order non-linear non-autonomous chaotic oscillator––part I

2004 ◽  
Vol 20 (4) ◽  
pp. 843-847 ◽  
Author(s):  
S.G. Stavrinides ◽  
K.G. Kyritsi ◽  
N.C. Deliolanis ◽  
A.N. Anagnostopoulos ◽  
A. Tamaševičious ◽  
...  
Keyword(s):  
Period Doubling ◽  
Second Order ◽  
Chaotic Oscillator ◽  
Non Linear ◽  
Route To Chaos

Sensitivity of routes to chaos in optically injected semiconductor lasers

2014 ◽  
Vol 23 (03) ◽  
pp. 1450036
Author(s):  
Najm M. Al-Hosiny
Keyword(s):  
Semiconductor Laser ◽  
Semiconductor Lasers ◽  
Period Doubling ◽  
Optical Signal ◽  
Optical Injection ◽  
Bifurcation Diagrams ◽  
Routes To Chaos ◽  
Route To Chaos

Two common routes to chaos, period-doubling and quasi-periodic, are theoretically investigated in semiconductor laser subject to optical injection. In particular, the sensitivity of the route to the injection of an additional optical signal is examined using bifurcation diagrams. Period-doubling route to chaos is found to be less sensitive to the perturbation of the second signal than the quasi-periodic route.

The use of Kripke's schema as a reduction principle

1977 ◽  
Vol 42 (2) ◽  
pp. 238-240 ◽  
Author(s):  
D. van Dalen
Keyword(s):  
Second Order ◽  
Reduction Technique ◽  
Reduction Principle ◽  
Natural Numbers ◽  
Second Order Arithmetic ◽  
Intuitionistic Theory ◽  
Order Arithmetic ◽  
Definition Of ◽  
Image Position ◽  
Creative Subject

The comprehension principle of second order arithmetic asserts the existence of certain species (sets) corresponding to properties of natural numbers. In the intuitionistic theory of sequences of natural numbers there is an analoguous principle, implicit in Brouwer's writing and explicitly stated by Kripke, which asserts the existence of certain sequences corresponding to statements. The justification of this principle, Kripke's Schema, makes use of the concept of the so-called creative subject. For more information the reader is referred to Troelstra [5].Kripke's Schema readsThere is a weaker versionThe idea to reduce species to sequences via Kripke's schema occurred several years ago (cf. [2, p. 128], [5, p. 104]). In [1] the reduction technique was applied in the construction of a model for HAS.On second thought, however, I realized that there is a straightforward, simpler way to exploit Kripke's schema, avoiding models altogether. The idea to present this material separately was forced on the author by C. Smorynski.Consider a second order arithmetic with both species and sequence variables. By KS we have(for convenience we restrict ourselves in KS to 0-1-sequences). An application of AC-NF givesOf course ξ is not uniquely determined. This is the key to the reduction of full second order arithmetic, or HAS, to a theory of sequences.We now introduce a translation τ to eliminate species variables. It is no restriction to suppose that the formulae contain only the sequence variables ξ1, ξ3, ξ5, …Note that by virtue of the definition of τ the axiom of extensionality is automatically verified after translation. The translation τ eliminates the species variables and leaves formulae without species variables invariant.

ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE

2004 ◽  
Vol 04 (01) ◽  
pp. 63-76 ◽  
Author(s):  
OLIVER JENKINSON
Keyword(s):  
Hausdorff Dimension ◽  
Finite Subset ◽  
Continued Fraction ◽  
Cantor Set ◽  
Computational Method ◽  
Accurate Approximation ◽  
Natural Numbers ◽  
Important Special Case ◽  
The One ◽  
Special Case

Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim (EA) is between 0 and 1. It is shown that the set [Formula: see text] intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim (EA), and employ it to investigate numerically the way in which [Formula: see text] intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbański, that [Formula: see text] is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim (E{1,2}), improving on the one given in [18].

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