Hopf Bifurcation of the Higher Dimensional 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)

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Complex Dynamics in Generalized Hénon Map

Discrete Dynamics in Nature and Society ◽

10.1155/2015/270604 ◽

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.

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The Generalized Hénon Map

Numerical Bifurcation Analysis of Maps ◽

10.1017/9781108585804.012 ◽

2019 ◽

pp. 321-353

Keyword(s):

Hénon Map ◽

Henon Map ◽

Generalized Hénon Map

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Time series classification based on triadic time series motifs

International Journal of Modern Physics B ◽

10.1142/s0217979219502370 ◽

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.

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Pseudo-random sequence generator based on the generalized Henon map

The Journal of China Universities of Posts and Telecommunications ◽

10.1016/s1005-8885(08)60109-0 ◽

2008 ◽

Vol 15 (3) ◽

pp. 64-68 ◽

Cited By ~ 33

Author(s):

Fan ZHENG ◽

Xiao-jian TIAN ◽

Jing-yi SONG ◽

Xue-yan LI

Keyword(s):

Random Sequence ◽

Hénon Map ◽

Henon Map ◽

Generalized Hénon Map ◽

Sequence Generator

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Hash function based on the generalized Henon map

Chinese Physics B ◽

10.1088/1674-1056/17/5/025 ◽

2008 ◽

Vol 17 (5) ◽

pp. 1685-1690 ◽

Cited By ~ 11

Author(s):

Zheng Fan ◽

Tian Xiao-Jian ◽

Li Xue-Yan ◽

Wu Bin

Keyword(s):

Hash Function ◽

Hénon Map ◽

Henon Map ◽

Generalized Hénon Map

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Transient chaos in a generalized Hénon map on the torus

Physics Letters A ◽

10.1016/0375-9601(88)90801-8 ◽

1988 ◽

Vol 126 (7) ◽

pp. 405-410 ◽

Cited By ~ 12

Author(s):

F.M. Izrailev ◽

B. Timmermann ◽

W. Timmermann

Keyword(s):

Transient Chaos ◽

Hénon Map ◽

Henon Map ◽

Generalized Hénon Map

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Hopf Bifurcation Analysis for a Stochastic Discrete-Time Hyperchaotic System

Discrete Dynamics in Nature and Society ◽

10.1155/2015/127404 ◽

2015 ◽

Vol 2015 ◽

pp. 1-12

Author(s):

Jie Ran ◽

Yu-Qin Li ◽

Shao-Juan Ma ◽

Juan Wu

Keyword(s):

Hopf Bifurcation ◽

Numerical Simulations ◽

Discrete Time ◽

Control Method ◽

Deterministic System ◽

Hyperchaotic System ◽

Discrete Time System ◽

Amplitude Control ◽

Random Disturbances ◽

Dynamical Features

The dynamics of a discrete-time hyperchaotic system and the amplitude control of Hopf bifurcation for a stochastic discrete-time hyperchaotic system are investigated in this paper. Numerical simulations are presented to exhibit the complex dynamical behaviors in the discrete-time hyperchaotic system. Furthermore, the stochastic discrete-time hyperchaotic system with random parameters is transformed into its equivalent deterministic system with the orthogonal polynomial theory of discrete random function. In addition, the dynamical features of the discrete-time hyperchaotic system with random disturbances are obtained through its equivalent deterministic system. By using the Hopf bifurcation conditions of the deterministic discrete-time system, the specific conditions for the existence of Hopf bifurcation in the equivalent deterministic system are derived. And the amplitude control with random intensity is discussed in detail. Finally, the feasibility of the control method is demonstrated by numerical simulations.

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