EXPERIMENTAL VERIFICATION FOR FRACTAL TRANSITION USING A FORCED DAMPED OSCILLATOR

Fractals ◽  
2000 ◽  
Vol 08 (01) ◽  
pp. 67-72 ◽  
Author(s):  
KAZUTOSHI GOHARA ◽  
HIROSHI SAKURAI ◽  
SHOZO SATO
Keyword(s):  
Correlation Dimension ◽  
Contraction Mappings ◽  
Fractal Sets ◽  
Self Similar ◽  
Initial States ◽  
Affine Set ◽  
Damped Oscillator ◽  
Set Of Initial States ◽  
Similarity Dimension ◽  
Good Agreement

A damped oscillator stochastically driven by temporal forces is experimentally investigated. The dynamics is characterized by a set Γ(C) of trajectories in a cylindrical space, where C is a set of initial states on the Poincaré section. Two sets, Γ(C) and C, are attractive and unique invariant fractal sets that approximately satisfy specific equations derived previously by the authors. The correlation dimension of the set C is in good agreement with the similarity dimension obtained for a strictly self-similar set constructed by contraction mappings while C is a self-affine set constructed by non-contraction mappings.

FRACTALS IN AN ELECTRONIC CIRCUIT WITH BY SWITCHING INPUTS

2002 ◽  
Vol 12 (04) ◽  
pp. 827-834 ◽  
Author(s):  
JUN NISHIKAWA ◽  
KAZUTOSHI GOHARA
Keyword(s):  
Correlation Dimension ◽  
Chaotic Dynamics ◽  
Electronic Circuit ◽  
Conventional Theory ◽  
Fractal Sets ◽  
Cylindrical Phase ◽  
Initial States ◽  
Set Of Initial States ◽  
Cylindrical Phase Space ◽  
Contraction Maps

We have proposed a process of generating fractals not from the results of chaotic dynamics, but from the switching of ordinary differential equations. This paper experimentally and numerically analyzes the dynamics of an electronic circuit driven by stochastically switching inputs. The following two results are obtained. First, the dynamics is characterized by a set Γ(C) of trajectories in the cylindrical phase space, where C is a set of initial states on the Poincaré section. Γ(C) and C are attractive and unique invariant fractal sets that satisfy specific equations. The second result is that the correlation dimension of C is in inverse proportion to the interval of the switching inputs. These two findings move beyond the conventional theory based on contraction maps. It should be noted that the set C is constructed by noncontraction maps.

scholarly journals SEPARATION PROPERTIES FOR ITERATED FUNCTION SYSTEMS OF BOUNDED DISTORTION

Fractals ◽  
2011 ◽  
Vol 19 (03) ◽  
pp. 259-269 ◽  
Author(s):  
MARIANO A. FERRARI ◽  
PABLO PANZONE
Keyword(s):  
Hausdorff Measure ◽  
Positive Measure ◽  
Iterated Function Systems ◽  
Separation Property ◽  
Fractal Sets ◽  
Iterated Function ◽  
Self Similar ◽  
Bounded Distortion ◽  
Weak Separation ◽  
Similarity Dimension

In this paper we study a general separation property for subsystems G, whose attractor KG is a sub-self-similar set. This is a generalization of the Lau-Ngai weak separation property for the bounded distortion case. For subsystems with positive Hausdorff measure in its similarity dimension, we characterize the subsets of KG with positive measure where the separation property may fail. We exhibit two examples of fractal sets, one not satisfying the weak separation property and whose existence was questioned by Zerner, the other having positive Hausdorff measure in its dimension and with the separation property failing on a subset of positive measure.

Falconer's formula for the Hausdorff dimension of a self-affine set in R2

1995 ◽  
Vol 15 (1) ◽  
pp. 77-97 ◽  
Author(s):  
Irene Hueter ◽  
Steven P. Lalley
Keyword(s):  
Hausdorff Dimension ◽  
Dimensional Hausdorff Measure ◽  
Similarity Transformations ◽  
Affine Mappings ◽  
Finite Set ◽  
Self Similar ◽  
Image Position ◽  
Affine Set ◽  
Small Overlap ◽  
Box Dimensions

Let A1, A2,…,Ak be a finite set of contractive, affine, invertible self-mappings of R2. A compact subset Λ of R2 is said to be self-affine with affinitiesA1, A2,…,Ak ifIt is known [8] that for any such set of contractive affine mappings there is a unique (compact) SA set with these affinities. When the affine mappings A1, A2,…,Ak are similarity transformations, the set Λ is said to be self-similar. Self-similar sets are well understood, at least when the images Ai(Λ) have ‘small’ overlap: there is a simple and explicit formula for the Hausdorff and box dimensions [12, 10]; these are always equal; and the δ-dimensional Hausdorff measure of such a set (where δ is the Hausdorff dimension) is always positive and finite.

Integrated wavelets on fractal sets. I. The correlation dimension

Nonlinearity ◽  
1992 ◽  
Vol 5 (3) ◽  
pp. 777-790 ◽  
Author(s):  
J -M Ghez ◽  
S Vaienti
Keyword(s):  
Correlation Dimension ◽  
Fractal Sets

KINEMATIC WAVE EQUATIONS FOR MOVABLE RIVERBEDS

2021 ◽  
pp. 43-54
Author(s):  
A. N. Krutov ◽  
◽  
S. Ya. Shkol’nikov ◽  
Keyword(s):  
Mathematical Model ◽  
Numerical Method ◽  
Wave Equations ◽  
Simple Wave ◽  
Kinematic Wave ◽  
Calculation Results ◽  
Self Similar ◽  
Similar Solutions ◽  
The Mathematical Model ◽  
Good Agreement

The mathematical model of kinematic wave, that is widely used in hydrological calculations, is generalized to compute processes in deformable channels. Self-similar solutions to the kinematic wave equations, namely, the discontinuous wave of increase and the “simple” wave of decrease are generalized. A numerical method is proposed for solving the kinematic wave equations for deformable channels. The comparison of calculation results with self-similar solutions revealed a good agreement.

scholarly journals Set Propagation Techniques for Reachability Analysis

2020 ◽  
Vol 4 (1) ◽  
Author(s):  
Matthias Althoff ◽  
Goran Frehse ◽  
Antoine Girard
Keyword(s):  
Hybrid Systems ◽  
Fundamental Problem ◽  
Autonomous Systems ◽  
Reachability Analysis ◽  
Controller Synthesis ◽  
Annual Review ◽  
Publication Date ◽  
Initial States ◽  
Set Of Initial States ◽  
Real World Problems

Reachability analysis consists in computing the set of states that are reachable by a dynamical system from all initial states and for all admissible inputs and parameters. It is a fundamental problem motivated by many applications in formal verification, controller synthesis, and estimation, to name only a few. This article focuses on a class of methods for computing a guaranteed overapproximation of the reachable set of continuous and hybrid systems, relying predominantly on set propagation; starting from the set of initial states, these techniques iteratively propagate a sequence of sets according to the system dynamics. After a review of set representation and computation, the article presents the state of the art of set propagation techniques for reachability analysis of linear, nonlinear, and hybrid systems. It ends with a discussion of successful applications of reachability analysis to real-world problems. Expected final online publication date for the Annual Review of Control, Robotics, and Autonomous Systems, Volume 4 is May 3, 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

scholarly journals Exceptional sets for self-similar fractals in Carnot groups

2010 ◽  
Vol 149 (1) ◽  
pp. 147-172 ◽  
Author(s):  
ZOLTÁN M. BALOGH ◽  
RETO BERGER ◽  
ROBERTO MONTI ◽  
JEREMY T. TYSON
Keyword(s):  
Hausdorff Dimension ◽  
Euclidean Space ◽  
Iterated Function Systems ◽  
Carnot Groups ◽  
Exceptional Set ◽  
Carathéodory Metric ◽  
Exceptional Sets ◽  
Iterated Function ◽  
Self Similar ◽  
Similarity Dimension

AbstractWe consider self-similar iterated function systems in the sub-Riemannian setting of Carnot groups. We estimate the Hausdorff dimension of the exceptional set of translation parameters for which the Hausdorff dimension in terms of the Carnot–Carathéodory metric is strictly less than the similarity dimension. This extends a recent result of Falconer and Miao from Euclidean space to Carnot groups.

Canonical number systems, counting automata and fractals

2002 ◽  
Vol 133 (1) ◽  
pp. 163-182 ◽  
Author(s):  
KLAUS SCHEICHER ◽  
JÖRG M. THUSWALDNER
Keyword(s):  
Direct Methods ◽  
Number Fields ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Box Counting ◽  
Number Systems ◽  
Rings Of Integers ◽  
Box Counting Dimension ◽  
Self Similar ◽  
Affine Set

In this paper we study properties of the fundamental domain [Fscr ]β of number systems, which are defined in rings of integers of number fields. First we construct addition automata for these number systems. Since [Fscr ]β defines a tiling of the n-dimensional vector space, we ask, which tiles of this tiling ‘touch’ [Fscr ]β. It turns out that the set of these tiles can be described with help of an automaton, which can be constructed via an easy algorithm which starts with the above-mentioned addition automaton. The addition automaton is also useful in order to determine the box counting dimension of the boundary of [Fscr ]β. Since this boundary is a so-called graph-directed self-affine set, it is not possible to apply the general theory for the calculation of the box counting dimension of self similar sets. Thus we have to use direct methods.

Research and Proof of Decidability on Self-Similar Fractals

2014 ◽  
Vol 511-512 ◽  
pp. 1185-1188
Author(s):  
Min Jin
Keyword(s):  
Hausdorff Dimension ◽  
Separation Condition ◽  
Strong Separation ◽  
Self Similar ◽  
Strong Separation Condition ◽  
Similarity Dimension

Some undecidability on self-affine fractals have been supported. In this paper, we research on the decidability for self-similar fractal of Dubes type. In fact, we prove that the following problems are decidable to test if the Hausdorff dimension of a given Dubes self-similar set is equal to its similarity dimension, and to test if a given Dubes self-similar set satisfies the strong separation condition.

scholarly journals Self-similar and self-affine sets: measure of the intersection of two copies

2009 ◽  
Vol 30 (2) ◽  
pp. 399-440 ◽  
Author(s):  
MÁRTON ELEKES ◽  
TAMÁS KELETI ◽  
ANDRÁS MÁTHÉ
Keyword(s):  
List Type ◽  
Small Perturbations ◽  
Invariant Extension ◽  
Affine Maps ◽  
And Topology ◽  
Self Similar ◽  
Affine Set ◽  
Separation Conditions

AbstractLetK⊂ℝdbe a self-similar or self-affine set and letμbe a self-similar or self-affine measure on it. Let 𝒢 be the group of affine maps, similitudes, isometries or translations of ℝd. Under various assumptions (such as separation conditions, or the assumption that the transformations are small perturbations, or thatKis a so-called Sierpiński sponge) we prove theorems of the following types, which are closely related to each other.•(Non-stability)There exists a constantc<1 such that for everyg∈𝒢 we have eitherμ(K∩g(K))<c⋅μ(K) orK⊂g(K).•(Measure and topology)For everyg∈𝒢 we haveμ(K∩g(K))>0⟺∫K(K∩g(K))≠0̸ (where ∫Kis interior relative toK).•(Extension)The measureμhas a 𝒢-invariant extension to ℝd.Moreover, in many situations we characterize thosegfor whichμ(K∩g(K))>0. We also obtain results about thosegfor whichg(K)⊂Korg(K)⊃K.

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