scholarly journals Complex Dynamics in Generalized Hénon Map

2015 ◽  
Vol 2015 ◽  
pp. 1-18
Author(s):  
Meixiang Cai
Keyword(s):  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Complex Dynamics ◽  
Chaotic Attractors ◽  
Jacobian Determinant ◽  
Flip Bifurcation ◽  
Hénon Map ◽  
Henon Map ◽  
Fold Bifurcation ◽  
Generalized Hénon Map

The complex dynamics of generalized Hénon map with nonconstant Jacobian determinant are investigated. The conditions of existence for fold bifurcation, flip bifurcation, and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory and checked up by numerical simulations. Chaos in the sense of Marotto's definition is proved by analytical and numerical methods. The numerical simulations show the consistence with the theoretical analysis and reveal some new complex phenomena which can not be given by theoretical analysis, such as the invariant cycles which are irregular closed graphics, the six and forty-one coexisting invariant cycles, and the two, six, seven, nine, ten, and thirteen coexisting chaotic attractors, and some kinds of strange chaotic attractors.

scholarly journals Hopf Bifurcation of the Higher Dimensional Hénon Map

2019 ◽  
Vol 67 (1) ◽  
pp. 73-78
Author(s):  
Saiful Islam ◽  
Chandra Nath Podder
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Discrete Time ◽  
Explicit Criterion ◽  
Hénon Map ◽  
Henon Map ◽  
Higher Dimensional ◽  
Generalized Hénon Map

In this paper, the discrete time generalized Hénon map is considered and the existence of Hopf bifurcation via an explicit criterion for N≥3, in particular for N=4 and N=5 has given. The relation between the parameters a and b as well as the range of the values of the parameters for N=3,4,5 has driven and the existence of Hopf bifurcation is demonstrated for the values of the parameters calculated from their relations. The results of numerical simulations for different values of the parameters are also presented. Dhaka Univ. J. Sci. 67(1): 73-78, 2019 (January)

scholarly journals A Novel 2D – Grid of Scroll Chaotic Attractor Generated by CNN

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 99 ◽  
Author(s):  
Ahmed M. Ali ◽  
Saif M. Ramadhan ◽  
Fadhil R. Tahir
Keyword(s):  
Theoretical Analysis ◽  
Chaotic Attractor ◽  
Complex Dynamics ◽  
Equilibrium Points ◽  
Chaotic Attractors ◽  
Electronic Circuits ◽  
Nonlinear Networks ◽  
Design And Implementation ◽  
Simulation Results ◽  
New System

The complex grid of scroll chaotic attractors that are generated through nonlinear electronic circuits have been raised considerably over the last decades. In this paper, it is shown that a subclass of Cellular Nonlinear Networks (CNNs) allows us to generate complex dynamics and chaos in symmetry pattern. A novel grid of scroll chaotic attractor, based on a new system, shows symmetry scrolls about the origin. Also, the equilibrium points are located in a manner such that the symmetry about the line x=y has been achieved. The complex dynamics of system can be generated using CNNs, which in turn are derived from a CNN array (1×3) cells. The paper concerns on the design and implementation of 2×2 and 3×3 2D-grid of scroll via the CNN model. Theoretical analysis and numerical simulations of the derived model are included. The simulation results reveal that the grid of scroll attractors can be successfully reproduced using PSpice.

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.

scholarly journals Time series classification based on triadic time series motifs

2019 ◽  
Vol 33 (21) ◽  
pp. 1950237
Author(s):  
Wen-Jie Xie ◽  
Rui-Qi Han ◽  
Wei-Xing Zhou
Keyword(s):  
Time Series ◽  
Logistic Map ◽  
Time Series Classification ◽  
Similarity Coefficients ◽  
Motif Analysis ◽  
Hénon Map ◽  
Henon Map ◽  
Motif Occurrence ◽  
Generalized Hénon Map ◽  
Dynamic Time

It is of great significance to identify the characteristics of time series to quantify their similarity and classify different classes of time series. We define six types of triadic time-series motifs and investigate the motif occurrence profiles extracted from the time series. Based on triadic time series motif profiles, we further propose to estimate the similarity coefficients between different time series and classify these time series with high accuracy. We validate the method with time series generated from nonlinear dynamic systems (logistic map, chaotic logistic map, chaotic Henon map, chaotic Ikeda map, hyperchaotic generalized Henon map and hyperchaotic folded-tower map) and retrieved from the UCR Time Series Classification Archive. Our analysis shows that the proposed triadic time series motif analysis performs better than the classic dynamic time wrapping method in classifying time series for certain datasets investigated in this work.

Hash function based on the generalized Henon map

2008 ◽  
Vol 17 (5) ◽  
pp. 1685-1690 ◽  
Author(s):  
Zheng Fan ◽  
Tian Xiao-Jian ◽  
Li Xue-Yan ◽  
Wu Bin
Keyword(s):  
Hash Function ◽  
Hénon Map ◽  
Henon Map ◽  
Generalized Hénon Map

Transient chaos in a generalized Hénon map on the torus

1988 ◽  
Vol 126 (7) ◽  
pp. 405-410 ◽  
Author(s):  
F.M. Izrailev ◽  
B. Timmermann ◽  
W. Timmermann
Keyword(s):  
Transient Chaos ◽  
Hénon Map ◽  
Henon Map ◽  
Generalized Hénon Map
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