Chaotic Dynamics in Two-Dimensional Noninvertible Maps

10.1142/2252 ◽  
1996 ◽  
Author(s):  
Christian Mira ◽  
Laura Gardini ◽  
Alexandra Barugola ◽  
Jean-Claude Cathala
Keyword(s):  
Chaotic Dynamics ◽  
Two Dimensional ◽  
Noninvertible Maps

BASIN BIFURCATIONS OF TWO-DIMENSIONAL NONINVERTIBLE MAPS: FRACTALIZATION OF BASINS

1994 ◽  
Vol 04 (02) ◽  
pp. 343-381 ◽  
Author(s):  
C. MIRA ◽  
D. FOURNIER-PRUNARET ◽  
L. GARDINI ◽  
H. KAWAKAMI ◽  
J.C. CATHALA
Keyword(s):  
Phase Plane ◽  
The Other ◽  
Two Dimensional ◽  
Basin Boundary ◽  
Critical Curves ◽  
Noninvertible Maps

Properties of the basins of noninvertible maps of a plane are studied using the method of critical curves. Different kinds of basin bifurcation, some of them leading to basin boundary fractalization are described. More particularly the paper considers the simplest class of maps that of a phase plane which is made up of two regions, one with two preimages, the other with no preimage.

Relation between Topological Entropies and Fractal Dimensions of Chaotic Attractors of the Hénon Map

2012 ◽  
Vol 569 ◽  
pp. 447-450
Author(s):  
Xiao Zhou Chen ◽  
Liang Lin Xiong ◽  
Long Li
Keyword(s):  
Linear Relation ◽  
Chaotic Dynamics ◽  
Fractal Dimensions ◽  
Chaotic Attractors ◽  
Two Dimensional ◽  
Hénon Map ◽  
Henon Map ◽  
Efficient Approach ◽  
Integral Relationship

In two-dimensional chaotic dynamics, relationship between fractal dimensions and topological entropies is an important issue to understand the chaotic attractors of Hénon map. we proposed a efficient approach for the estimation of topological entropies through the study on the integral relationship between fractal dimensions and topological entropies. Our result found that there is an approximate linear relation between their topological entropies and fractal dimensions.

FRACTAL AGGREGATION OF BASIN ISLANDS IN TWO-DIMENSIONAL QUADRATIC NONINVERTIBLE MAPS

1995 ◽  
Vol 05 (04) ◽  
pp. 991-1019 ◽  
Author(s):  
C. MIRA ◽  
C. RAUZY
Keyword(s):  
Phase Plane ◽  
Parameter Variation ◽  
The Other ◽  
Two Dimensional ◽  
Quadratic Maps ◽  
Attracting Set ◽  
Critical Curves ◽  
Noninvertible Maps

Properties of basins of noninvertible maps of the plane are studied by using the method of critical curves. The paper considers the simplest class of quadratic maps, that having a phase plane made up of two regions, one with two first rank preimages, the other with no preimage, in situations different from those described in a previous publication. More specifically, the considered quadratic maps give rise to a basin made up of infinitely many nonconnected regions, a parameter variation leading to an aggregation of these regions, which occur in a fractal way. The nonconnected regions, different from that containing an attracting set, are called "islands".

GENERATION OF SYMMETRICAL COLORED IMAGES VIA SOLUTION OF THE INVERSE PROBLEM OF CHEMICAL REACTIONS DISCRETE CHAOTIC DYNAMICS

2006 ◽  
Vol 16 (05) ◽  
pp. 1419-1434 ◽  
Author(s):  
V. GONTAR ◽  
O. GRECHKO
Keyword(s):  
Genetic Algorithm ◽  
Mathematical Model ◽  
Inverse Problem ◽  
Mathematical Models ◽  
Chemical Reactions ◽  
Chaotic Dynamics ◽  
Two Dimensional

An automatic procedure for generating colored two-dimensional symmetrical images based on the chemical reactions discrete chaotic dynamics (CRDCD) is proposed. The inverse problem of derivation of symmetrical images from CRDCD mathematical models was formulated and solved using a special type of genetic algorithm. Different symmetrical images corresponding to the solutions of a CRDCD mathematical model for which the parameters were obtained automatically by the proposed method are presented.

Homoclinic and Heteroclinic Situations Specific to Two-Dimensional Noninvertible Maps

1997 ◽  
Vol 07 (01) ◽  
pp. 39-70 ◽  
Author(s):  
Gilles Millerioux ◽  
Christian Mira
Keyword(s):  
Qualitative Change ◽  
Closed Curve ◽  
Stable Set ◽  
Complex Structures ◽  
Two Dimensional ◽  
Parameter Region ◽  
Critical Curves ◽  
Rank One ◽  
Stable And Unstable Sets ◽  
Noninvertible Maps

These situations are put in evidence from two examples of (Z0 - Z2) maps. It is recalled that such maps (the simplest type of non-invertible ones) are related to the separation of the plane into a region without preimage, and a region each point of which has two rank-one preimages. With respect to diffeomorphisms, non-invertibility introduces more complex structures of the stable and unstable sets defining the homoclinic and heteroclinic situations, and the corresponding bifurcations. Critical curves permit the analysis of this question. More particularly, a basic global contact bifurcation (contact of the map critical curve with a non-connected saddle stable set Ws) plays a fundamental role for explaining the qualitative change of Ws, which occurs between two boundary homoclinic bifurcations limiting a parameter region related to the disappearing of an attracting invariant closed curve.

A Solution for Two-Dimensional Mazes with Use of Chaotic Dynamics in a Recurrent Neural Network Model

2004 ◽  
Vol 16 (9) ◽  
pp. 1943-1957 ◽  
Author(s):  
Yoshikazu Suemitsu ◽  
Shigetoshi Nara
Keyword(s):  
Neural Network ◽  
State Space ◽  
Network Model ◽  
Neural Network Model ◽  
Chaotic Dynamics ◽  
Dimensional Space ◽  
Dynamical Structure ◽  
Two Dimensional ◽  
Time Step ◽  
The Neural Network

Chaotic dynamics introduced into a neural network model is applied to solving two-dimensional mazes, which are ill-posed problems. A moving object moves from the position at t to t + 1 by simply defined motion function calculated from firing patterns of the neural network model at each time step t. We have embedded several prototype attractors that correspond to the simple motion of the object orienting toward several directions in two-dimensional space in our neural network model. Introducing chaotic dynamics into the network gives outputs sampled from intermediate state points between embedded attractors in a state space, and these dynamics enable the object to move in various directions. System parameter switching between a chaotic and an attractor regime in the state space of the neural network enables the object to move to a set target in a two-dimensional maze. Results of computer simulations show that the success rate for this method over 300 trials is higher than that of random walk. To investigate why the proposed method gives better performance, we calculate and discuss statistical data with respect to dynamical structure.

Chaotic dynamics for two-dimensional tent maps

Nonlinearity ◽  
2015 ◽  
Vol 28 (2) ◽  
pp. 407-434 ◽  
Author(s):  
Antonio Pumariño ◽  
José Ángel Rodríguez ◽  
Joan Carles Tatjer ◽  
Enrique Vigil
Keyword(s):  
Chaotic Dynamics ◽  
Two Dimensional ◽  
Tent Maps
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