A functional equation for a segment of the Hénon map unstable manifold

1986 ◽  
Vol 21 (2-3) ◽  
pp. 415-426 ◽  
Author(s):  
A. Boyarsky
Keyword(s):  
Functional Equation ◽  
Unstable Manifold ◽  
Hénon Map ◽  
Henon Map

Unstable Manifold of Hénon Map

2014 ◽  
Vol 12 (2) ◽  
Author(s):  
LI Zhongqin ◽  
JIA Meng
Keyword(s):  
Unstable Manifold ◽  
Hénon Map ◽  
Henon Map

Horseshoes in the measure-preserving Hénon map

1995 ◽  
Vol 15 (6) ◽  
pp. 1045-1059 ◽  
Author(s):  
Ray Brown
Keyword(s):  
Fixed Point ◽  
Unstable Manifold ◽  
Recursive Formula ◽  
Homoclinic Point ◽  
Hénon Map ◽  
Henon Map ◽  
Broad Array ◽  
Hyperbolic Fixed Point ◽  
Elementary Methods

AbstractWe show, using elementary methods, that for 0 < a the measure-preserving, orientation-preserving Hénon map, H, has a horseshoe. This improves on the result of Devaney and Nitecki who have shown that a horseshoe exists in this map for a ≥ 8. For a > 0, we also prove the conjecture of Devaney that the first symmetric homoclinic point is transversal.To obtain our results, we show that for a branch, Cu, of the unstable manifold of a hyperbolic fixed point of H, Cu crosses the line y = − x and that this crossing is a homoclinic point, χc. This has been shown by Devaney, but we obtain the crossing using simpler methods. Next we show that if the crossing of Wu(p) and Ws(p) at χc is degenerate then the slope of Cu at this crossing is one. Following this we show that if χc is a degenerate homoclinic its x-coordinate must be greater than l/(2a). We then derive a contradiction from this by showing that the slope of Cu at H-1(χc) must be both positive and negative, thus we conclude that χc is transversal.Our approach uses a lemma that gives a recursive formula for the sign of curvature of the unstable manifold. This lemma, referred to as ‘the curvature lemma’, is the key to reducing the proof to elementary methods. A curvature lemma can be derived for a very broad array of maps making the applicability of these methods very general. Further, since curvature is the strongest differentiability feature needed in our proof, the methods work for maps of the plane which are only C2.

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.

Lots of Chaotic Behaviors of the Henon Map

2016 ◽  
Vol 12 (36) ◽  
pp. 411
Author(s):  
Md. Shakhawat Alam ◽  
Payer Ahmed
Keyword(s):  
Functional Equation ◽  
Fixed Points ◽  
Unique Solution ◽  
Local Stability ◽  
Continuous Dependence ◽  
Periodic Points ◽  
Topological Conjugacy ◽  
Hénon Map ◽  
Henon Map

In this paper we are concerned with a general form of the Henon map as a retarded functional equation. The existence of a unique solution is proved. The continuous dependence of the solution and the local stability of fixed points are investigated. Dynamics of periodic points, Chaos, bifurcation, and topological conjugacy of the resulting system are discussed in Matlab.

Double color image encryption based on fractional order discrete improved Henon map and Rubik’s cube transform

2021 ◽  
pp. 116363
Author(s):  
Liping Chen ◽  
Yin Hao ◽  
Liguo Yuan ◽  
J.A. Tenreiro Machado ◽  
Ranchao Wu ◽  
...  
Keyword(s):  
Image Encryption ◽  
Fractional Order ◽  
Color Image ◽  
Hénon Map ◽  
Color Image Encryption ◽  
Henon Map ◽  
Rubik's Cube ◽  
Rubik’S Cube

scholarly journals A transcendental Hénon map with an oscillating wandering Short $${\mathbb {C}}^2$$

2021 ◽  
Author(s):  
Leandro Arosio ◽  
Luka Boc Thaler ◽  
Han Peters
Keyword(s):  
Hénon Map ◽  
Henon Map

Collaboration of the Radial Basis ART and PSO in Multi-Solution Problems of the Hénon Map

2014 ◽  
pp. 221-228
Author(s):  
Fumiaki Tokunaga ◽  
Takumi Sato ◽  
Toshimichi Saito
Keyword(s):  
Hénon Map ◽  
Henon Map ◽  
Radial Basis

Autoregressive self-tuning feedback control of the Hénon map

1996 ◽  
Vol 54 (6) ◽  
pp. 6201-6206 ◽  
Author(s):  
Michael E. Brandt ◽  
Ahmet Ademoǧlu ◽  
Dejian Lai ◽  
Guanrong Chen
Keyword(s):  
Feedback Control ◽  
Hénon Map ◽  
Henon Map ◽  
Self Tuning

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.

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