EXTENSIONS OF THE NOTION OF CHAOTIC AREA IN SECOND-ORDER ENDOMORPHISMS

1995 ◽  
Vol 05 (03) ◽  
pp. 751-777 ◽  
Author(s):  
A. BARUGOLA ◽  
J.C. CATHALA ◽  
C. MIRA
Keyword(s):  
Fixed Point ◽  
Finite Number ◽  
Periodic Point ◽  
Critical Points ◽  
Unstable Manifold ◽  
Two Dimensional ◽  
One Dimensional ◽  
Critical Curves ◽  
The One ◽  
Invariant Domains

Properties of chaotic areas (i.e. invariant domains of points positively stable in the Poisson’s sense) of non-invertible maps of the plane are studied by using the method of critical curves (two-dimensional extension of the notion of critical points in the one-dimensional case). The classical situation is that of a chaotic area bounded by a finite number of critical curves segments. This paper considers another class of chaotic areas bounded by the union of critical curves segments and segments of the unstable manifold of a saddle fixed point, or that of saddle cycle (periodic point). Different configurations are examined, as their bifurcations when a map parameter varies.

ON BEHAVIORS OF TWO-DIMENSIONAL ENDOMORPHISMS: ROLE OF THE CRITICAL CURVES

1993 ◽  
Vol 03 (01) ◽  
pp. 187-194 ◽  
Author(s):  
CHRISTIAN MIRA ◽  
TONY NARAYANINSAMY
Keyword(s):  
Critical Points ◽  
Two Dimensional ◽  
One Dimensional ◽  
Critical Curves ◽  
Dynamical Properties

Critical curves are the natural two-dimensional extension of the notion of critical points in one-dimensional endomorphisms. They play a fundamental role in determining the dynamical properties and their bifurcations. This letter demonstrates such a role for two new behaviors.

TWO-DIMENSIONAL COUPLED PARAMETRICALLY FORCED MAP

2013 ◽  
Vol 23 (02) ◽  
pp. 1350031 ◽  
Author(s):  
HIRONORI KUMENO ◽  
DANIÈLE FOURNIER-PRUNARET ◽  
ABDEL-KADDOUS TAHA ◽  
YOSHIFUMI NISHIO
Keyword(s):  
Newton Method ◽  
Saddle Points ◽  
Logistic Map ◽  
Two Dimensional ◽  
Bifurcation Structure ◽  
One Dimensional ◽  
Stable Order ◽  
Critical Curves ◽  
Cusp Points ◽  
The One

A two-dimensional parametrically forced system constructed from two identical one-dimensional subsystems, whose parameters are forced into periodic varying, with mutually influencing coupling is proposed. We investigate bifurcations and basins in the parametrically forced system when logistic map is used for the one-dimensional subsystem. On a parameter plane, crossroad areas centered at fold cusp points for several orders are detected. From the investigation, a foliated bifurcation structure is drawn, and existence domains of stable order cycles with synchronization or without synchronization are detected. Moreover, evolution of bifurcation curves with respect to a coupling intensity is analyzed. Basin bifurcations and preimages with respect to critical curves are described. Basins where boundary depends on the invariant manifold of saddle points are numerically analyzed by considering second order iteration and using superposition with Newton method, although the system has discontinuity regarding parameters.

FPGA Implementations for Chaotic Maps Using Fixed-Point and Floating-Point Representations

2016 ◽  
pp. 59-97
Author(s):  
Ricardo Francisco Martinez-Gonzalez ◽  
Ruben Vazquez-Medina ◽  
Jose Alejandro Diaz-Mendez ◽  
Juan Lopez-Hernandez
Keyword(s):  
Fixed Point ◽  
Mathematical Description ◽  
Chaotic Maps ◽  
Similarity Coefficient ◽  
Floating Point ◽  
Two Dimensional ◽  
One Dimensional ◽  
The One ◽  
One Dimensional Maps

This work presents the implementation of various chaotic maps; among the maps there are one-dimensional and two-dimensional ones. In order to implement the maps, their mathematical descriptions are modified to be represented with more accuracy by different binary representations. The sequences from the same map are compared to determine until which iteration, different descriptions produce similar outputs. The similarity coefficient is established in five percent. Comparison delivers some interesting findings; first, the one-dimensional maps, in this work, have comparative number of similar iterations. Second, the bi-dimensional maps present the lowest and highest number of similar iterations. Based on the modified mathematical descriptions, the VHDL implementations are developed. They are simulated and their results are compared against the modified mathematical description ones; resulting that both groups of results are congruent.

scholarly journals Nilpotency and periodic points in non-uniform cellular automata

2021 ◽  
Vol 58 (4) ◽  
pp. 319-333
Author(s):  
Supreeti Kamilya ◽  
Jarkko Kari
Keyword(s):  
Fixed Point ◽  
Cellular Automata ◽  
Finite Number ◽  
Periodic Point ◽  
Unique Fixed Point ◽  
Periodic Points ◽  
One Dimensional ◽  
Periodic Distribution ◽  
Wang Tile

AbstractNilpotent cellular automata have the simplest possible dynamics: all initial configurations lead in bounded time into the unique fixed point of the system. We investigate nilpotency in the setup of one-dimensional non-uniform cellular automata (NUCA) where different cells may use different local rules. There are infinitely many cells in NUCA but only a finite number of different local rules. Changing the distribution of the local rules in the system may drastically change the dynamics. We prove that if the available local rules are such that every periodic distribution of the rules leads to nilpotent behavior then so do also all eventually periodic distributions. However, in some cases there may be non-periodic distributions that are not nilpotent even if all periodic distributions are nilpotent. We demonstrate such a possibility using aperiodic Wang tile sets. We also investigate temporally periodic points in NUCA. In contrast to classical uniform cellular automata, there are NUCA—even reversible equicontinuous ones—that do not have any temporally periodic points. We prove the undecidability of this property: there is no algorithm to determine if a NUCA with a given finite distribution of local rules has a periodic point.

SOME PROPERTIES OF A TWO-DIMENSIONAL PIECEWISE-LINEAR NONINVERTIBLE MAP

1996 ◽  
Vol 06 (12a) ◽  
pp. 2299-2319 ◽  
Author(s):  
CHRISTIAN MIRA ◽  
CHRISTINE RAUZY ◽  
YURI MAISTRENKO ◽  
IRINA SUSHKO
Keyword(s):  
Fixed Points ◽  
Piecewise Linear ◽  
Two Dimensional ◽  
One Dimensional ◽  
Critical Curves ◽  
Unstable Set ◽  
Rank One ◽  
Characteristic Features ◽  
The One ◽  
Parameter Values

Properties of a piecewise-linear noninvertible map of the plane are studied by using the method of critical curves (two-dimensional extension of the notion of critical point in the one-dimensional case). This map is of (Z0–Z2) type, i.e. the plane consists of a region without preimage, and a region giving rise to two rank one preimages. For the considered parameter values, the map has two saddle fixed points. The characteristic features of the “mixed chaotic area” generated by this map, and its bifurcations (some of them being of homoclinic and heteroclinic type) are examined. Such an area is bounded by the union of critical curves segments and segments of the unstable set of saddle cycles.

Regions-based Two-dimensional Continua: The Euclidean Case

2018 ◽  
Author(s):  
Geoffrey Hellman ◽  
Stewart Shapiro
Keyword(s):  
Mathematical Induction ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Entire Space ◽  
Euclidean Case ◽  
Free Basis ◽  
One Dimensional ◽  
Archimedean Property ◽  
The One

This chapter develops a Euclidean, two-dimensional, regions-based theory. As with the semi-Aristotelian account in Chapter 2, the goal here is to recover the now orthodox Dedekind–Cantor continuum on a point-free basis. The chapter derives the Archimedean property for a class of readily postulated orientations of certain special regions, what are called “generalized quadrilaterals” (intended as parallelograms), by which the entire space is covered. Then the chapter generalizes this to arbitrary orientations, and then establishes an isomorphism between the space and the usual point-based one. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit “extremal clause”, and we have no axiom of induction other than ordinary numerical (mathematical) induction.

MUTUAL EXCLUSION ON OPTICAL BUSES

2002 ◽  
Vol 12 (03n04) ◽  
pp. 341-358
Author(s):  
KRISHNA M. KAVI ◽  
DINESH P. MEHTA
Keyword(s):  
Mutual Exclusion ◽  
Two Dimensional ◽  
One Dimensional ◽  
Dimensional Array ◽  
Propagation Delays ◽  
Optical Buses ◽  
The One

This paper presents two algorithms for mutual exclusion on optical bus architectures including the folded one-dimensional bus, the one-dimensional array with pipelined buses (1D APPB), and the two-dimensional array with pipelined buses (2D APPB). The first algorithm guarantees mutual exclusion, while the second guarantees both mutual exclusion and fairness. Both algorithms exploit the predictability of propagation delays in optical buses.

scholarly journals DIFFERENTIAL EQUATION APPROACH FOR ONE- AND TWO-DIMENSIONAL LATTICE GREEN'S FUNCTION

2007 ◽  
Vol 21 (02n03) ◽  
pp. 139-154 ◽  
Author(s):  
J. H. ASAD
Keyword(s):  
Differential Equation ◽  
Green’S Function ◽  
Recurrence Relation ◽  
Green's Function ◽  
Two Dimensional ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
The One ◽  
Lattice Green's Function ◽  
Lattice Green’S Function

A first-order differential equation of Green's function, at the origin G(0), for the one-dimensional lattice is derived by simple recurrence relation. Green's function at site (m) is then calculated in terms of G(0). A simple recurrence relation connecting the lattice Green's function at the site (m, n) and the first derivative of the lattice Green's function at the site (m ± 1, n) is presented for the two-dimensional lattice, a differential equation of second order in G(0, 0) is obtained. By making use of the latter recurrence relation, lattice Green's function at an arbitrary site is obtained in closed form. Finally, the phase shift and scattering cross-section are evaluated analytically and numerically for one- and two-impurities.

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