scholarly journals An Algorithm to Automatically Detect the Smale Horseshoes

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Qingdu Li ◽  
Lina Zhang ◽  
Fangyan Yang
Keyword(s):  
Dynamical Systems ◽  
Chaotic Maps ◽  
Two Dimensional ◽  
Hénon Map ◽  
Henon Map ◽  
Unstable Manifolds ◽  
Duffing Map

Smale horseshoes, curvilinear rectangles and their U-shaped images patterned on Smale's famous example, provide a rigorous way to study chaos in dynamical systems. The paper is devoted to constructing them in two-dimensional diffeomorphisms with the existence of transversal homoclinic saddles. We first propose an algorithm to automatically construct “horizontal” and “vertical” sides of the curvilinear rectangle near to segments of the stable and of the unstable manifolds, respectively, and then apply it to four classical chaotic maps (the Duffing map, the Hénon map, the Ikeda map, and the Lozi map) to verify its effectiveness.

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.

Bifurcations of one- and two-dimensional maps

1984 ◽  
Vol 311 (1515) ◽  
pp. 43-102 ◽  
Keyword(s):  
Periodic Orbits ◽  
Period Doubling ◽  
Jacobian Determinant ◽  
Two Dimensional ◽  
Hénon Map ◽  
Stable And Unstable Manifolds ◽  
Henon Map ◽  
One Dimensional ◽  
The Family ◽  
Area Preserving

We study the qualitative dynamics of two-parameter families of planar maps of the form F^e(x, y) = (y, -ex+f(y)), where f :R -> R is a C 3 map with a single critical point and negative Schwarzian derivative. The prototype of such maps is the family f(y) = u —y 2 or (in different coordinates) f(y) = Ay(1 —y), in which case F^ e is the Henon map. The maps F e have constant Jacobian determinant e and, as e -> 0, collapse to the family f^. The behaviour of such one-dimensional families is quite well understood, and we are able to use their bifurcation structures and information on their non-wandering sets to obtain results on both local and global bifurcations of F/ ue , for small e . Moreover, we are able to extend these results to the area preserving family F/u. 1 , thereby obtaining (partial) bifurcation sets in the (/u, e)-plane. Among our conclusions we find that the bifurcation sequence for periodic orbits, which is restricted by Sarkovskii’s theorem and the kneading theory for one-dimensional maps, is quite different for two-dimensional families. In particular, certain periodic orbits that appear at the end of the one-dimensional sequence appear at the beginning of the area preserving sequence, and infinitely many families of saddle node and period doubling bifurcation curves cross each other in the ( /u, e ) -parameter plane between e = 0 and e = 1. We obtain these results from a study of the homoclinic bifurcations (tangencies of stable and unstable manifolds) of F /u.e and of the associated sequences of periodic bifurcations that accumulate on them. We illustrate our results with some numerical computations for the orientation-preserving Henon map.

THE ROLE OF CODIMENSION IN DYNAMICAL SYSTEMS

1992 ◽  
Vol 03 (06) ◽  
pp. 1295-1321 ◽  
Author(s):  
JASON A.C. GALLAS
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Period Doubling ◽  
Main Body ◽  
Hénon Map ◽  
Henon Map ◽  
Dynamical Parameters

Isoperiodic diagrams are used to investigate the topology of the codimension space of a representative dynamical system: the Hénon map. The codimension space is reported to be organized in a simple and regular way: instead of “structures-within-structures” it consists of a “structures-parallel-to-structures” sequence of shrimp-shaped isoperiodic islands immersed on a via caotica. The isoperiodic islands consist of a main body of principal periodicity k=1, 2, 3, 4, …, which bifurcates according to a period-doubling route. The Pk=k×2n, n=0, 1, 2, … shrimps are very densely concentrated along a main α-direction, a neighborhood parallel to the line b=−0.583a+1.025, where a and b are the dynamical parameters in Eq. (1). Isoperiodic diagrams allow to interpret and unify some apparently uncorrelated phenomena, such as ‘period-bubbling’, classes of reverse bifurcations and antimonotonicity and to recognize that they are in fact signatures of the complicated way in which period-doubling occurs in higher codimensional systems.

scholarly journals A novel multi-image cryptosystem based on weighted plain images and using combined chaotic maps

2021 ◽  
Author(s):  
Ahmad Pourjabbar Kari ◽  
Ahmad Habibizad Navin ◽  
Amir Massoud Bidgoli ◽  
Mirkamal Mirnia
Keyword(s):  
Chaotic Maps ◽  
The Other ◽  
Second Step ◽  
Hénon Map ◽  
Henon Map ◽  
Chaotic Sequences ◽  
Xor Operation ◽  
Plain Image ◽  
Diffusion Phase ◽  
Image Cryptosystem

Abstract This paper introduces a new multi-image cryptosystem based on modified Henon map and nonlinear combination of chaotic seed maps. Based on the degree of correlation between the adjacent pixels of the plain image, a unique weight is assigned to the plain image. First, the coordinates of plain images are disrupted by modified Henon map as confusion phase. In the first step of diffusion phase, the pixels content of images are changed separately by XOR operation between confused images and matrices with suitable nonlinear combination of seed maps sequences. These combination of seed maps are selected depending on the weight of plain images as well as bifurcation properties of mentioned chaotic maps. After concatenating the matrices obtained from the first step of diffusion phase, the bitwise XOR operation is applied between newly developed matrix and the other produced matrix from the chaotic sequences of the Logistic-Tent-Sine hybrid system, as second step of diffusion phase. The encrypted image is obtained after applying shift and exchange operations. The results of the implementation using graphs and histograms show that the proposed scheme, compared to some existing methods, can effectively resist common attacks and can be used as a secure method for encrypting digital images.

On Stable and Unstable Periodic Solutions of N-Dimensional Discrete Dynamical Systems

2009 ◽  
Author(s):  
Albert C. J. Luo ◽  
Yu Guo
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Discrete System ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Hénon Map ◽  
Henon Map ◽  
Discrete Maps ◽  
Mapping Structures ◽  
Bifurcation And Stability

This paper presents a methodology to analytically predict the stable and unstable periodic solutions for n-dimensional discrete dynamical systems. The positive and negative iterative mappings of discrete maps are introduced for the mapping structure of the periodic solutions. The complete bifurcation and stability of the stable and unstable periodic solutions relative to the positive and negative mapping structures are presented. A discrete dynamical system with the Henon map is investigated as an example. The Poincare mapping sections relative to the Neimark bifurcation of periodic solutions are presented, and the chaotic layers for the discrete system with the Henon map are observed.

Digital Henon Sequences Generation and its Analysis

2014 ◽  
Vol 981 ◽  
pp. 793-796 ◽  
Author(s):  
Bing Bing Song ◽  
Jing Pan ◽  
Qun Ding
Keyword(s):  
Numerical Results ◽  
Statistical Properties ◽  
Two Dimensional ◽  
Hénon Map ◽  
Digital Output ◽  
Balance Test ◽  
Henon Map ◽  
Output Sequence ◽  
Dsp Builder ◽  
Run Test

In this paper, the typical two-dimensional Henon map is studied. Firstly, the model of Henon map is proposed based on DSP Builder platform in Simulink library, so it can generate digital output sequence of Henon map. Then, its statistical properties are analyzed for such output sequences, including balance test, run test and autocorrelation test. Finally, the numerical results show that such digital Henon sequences have good pseudo-randomness.

Unstable Manifold of Hénon Map

2012 ◽  
Vol 569 ◽  
pp. 818-821
Author(s):  
Bo Chen ◽  
Meng Jia
Keyword(s):  
Unstable Manifold ◽  
The Other ◽  
Local Curvature ◽  
Hénon Map ◽  
Henon Map ◽  
One Dimensional ◽  
Iteration Sequence ◽  
Direct Iteration ◽  
Unstable Manifolds

A new algorithm is presented for computing one dimensional unstable manifold of a map and Hénon map is taken as an example to test the performance of the algorithm. The unstable manifold is grown with new point added at each step and the distance between consecutive points is adjusted according to the local curvature. It is proved that the gradient of the manifold at the new point can be predicted by the known points on the manifold and in this way the preimage of the new point could be located immediately. During the simulation, it is found that the unstable manifold of Hénon map coincides with its direct iteration when canonical parameters are chosen which means order is obtained out of chaos. In the other several groups of parameters the two branches of the unstable manifolds are nearly symmetric, and they serve as the borderline of the Hénon map iteration sequence. We hope that this would contribute to the further exploration of Hénon map.

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