Gradient Control of Hènon Map Dynamics

1997 ◽  
Vol 07 (03) ◽  
pp. 701-705 ◽  
Author(s):  
Peter Yu. Guzenko ◽  
Alexander L. Fradkov
Keyword(s):  
Discrete System ◽  
Complex Dynamics ◽  
System Stability ◽  
Two Dimensional ◽  
Hénon Map ◽  
Henon Map ◽  
Model Reference Control ◽  
Gradient Based ◽  
Gradient Control ◽  
Simulation Results

The paper is devoted to gradient control of the Hènon map — a simple two-dimensional discrete system with complex dynamics. Some gradient-based algorithms are suggested for stabilizing fixed point and model-reference control. Some sufficient system stability conditions are obtained and discussed. Simulation results demonstrating the performance of the suggested algorithms also included for the purpose of illustration.

scholarly journals Detecting and stabilizing periodic orbits of chaotic Henon map through predictive control

2018 ◽  
Vol 27 (2018) ◽  
pp. 73-78
Author(s):  
Dumitru Deleanu
Keyword(s):  
Periodic Orbits ◽  
Predictive Control ◽  
Nonnegative Integer ◽  
Strange Attractor ◽  
Discrete System ◽  
Control Method ◽  
Nonlinear Mapping ◽  
Unstable Periodic Orbits ◽  
Hénon Map ◽  
Henon Map

The predictive control method is one of the proposed techniques based on the location and stabilization of the unstable periodic orbits (UPOs) embedded in the strange attractor of a nonlinear mapping. It assumes the addition of a small control term to the uncontrolled state of the discrete system. This term depends on the predictive state ps + 1 and p(s + 1) + 1 iterations forward, where s is the length of the UPO, and p is a large enough nonnegative integer. In this paper, extensive numerical simulations on the Henon map are carried out to confirm the ability of the predictive control to detect and stabilize all the UPOs up to a maximum length of the period. The role played by each involved parameter is investigated and additional results to those reported in the literature are presented.

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.

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.

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.

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