From the Hénon conservative map to the Chirikov standard map for large parameter values

In this paper we consider conservative quadratic Hénon maps and Chirikov’s standard map, and relate them in some sense. First, we present a study of some dynamical properties of orientation-preserving and orientation-reversing quadratic Hénon maps concerning the stability region, the size of the chaotic zones, its evolution with respect to parameters and the splitting of the separatrices of fixed and periodic points plus its role in the preceding aspects. Then the phase space of the standard map, for large values of the parameter k, is studied. There are some stable orbits which appear periodically in k and are scaled somehow. Using this scaling, we show that the dynamics around these stable orbits is one of the above Hénon maps plus some small error, which tends to vanish as k→∞. Elementary considerations about diffusion properties of the standard map are also presented.     

On computing Poincaré map by Hénon method

The trajectory of the autonomous chaotic system deviates from the original path leading to a deformation in its attractor while calculating Poincaré map using the method presented by Hénon [Hénon M. Physica D 1982;5:412]. Also, the Poincaré map obtained is found to be the Poincaré map of deformed attractor instead of the original attractor. In order to overcome these drawbacks, this method is slightly modified by introducing an important change in the existing algorithm. Then it is shown that the modified Hénon method calculates the Poincaré map of the original attractor and it does not affect the system dynamics (attractor). The modified method is illustrated by means of the Lorenz and Chua systems.     

A note on homoclinic or heteroclinic orbits for the generalized Hénon map

An important problem in a given dynamical system is to determine the existence of a homoclinic orbit. We improve the results of Qin and Xiao [Nonlinearity, 20 (2007), 2305–2317], who present some sufficient conditions for the existence of a homoclinic/heteroclinic orbit for the generalized H´enon map. Moreover, an algorithm is presented to locate these homoclinic orbits.     

Elliptic solutions in the Hénon–Heiles model

Equations of motion corresponding to the Hénon–Heiles Hamiltonian are considered. A method enabling one to find all elliptic solutions of an autonomous ordinary differential equation or a system of autonomous ordinary differential equations is described. New families of elliptic solutions of a fourth-order equation related to the Hénon–Heiles system are obtained. A classification of elliptic solutions up to the sixth order inclusively is presented.     

Chaotic control of Hénon map with feedback and nonfeedback methods

In this paper, based on the stability theorem of discrete systems and the idea of less energy, which means more stability in physical systems, feedback and nonfeedback methods are used, respectively to stabilize chaotic Hénon map at different p-periodic orbits. The two methods are compared in different aspects. Numerical simulations show the feasibility and effectiveness of them.     

Entropy estimation of the Hénon attractor

The topological entropy of the Hénon attractor is estimated using a function that describes the stable and unstable manifolds of the Hénon map. This function provides an accurate estimate of the length of curves in the attractor. The estimation method presented here can be applied to cases in which the invariant set is not hyperbolic. From the result of the length calculation, we have estimated the topological entropy h as h ～ 0.49703 for the original parameters a = 1.4 and b = 0.3 adopted by Hénon.     

Renormalization in the Hénon family,II: the heteroclinic web

We study highly dissipative Hénon maps

$F_{c,b}: (x,y) \mapsto (c-x^2-by, x)$

with zero entropy. They form a region Π in the parameter plane bounded on the left by the curve W of infinitely renormalizable maps. We prove that Morse-Smale maps are dense in Π, but there exist infinitely many different topological types of such maps (even away from W). We also prove that in the infinitely renormalizable case, the average Jacobian b _F on the attracting Cantor set \({\mathcal{O}}_{F}\) is a topological invariant. These results come from the analysis of the heteroclinic web of the saddle periodic points based on the renormalization theory. Along these lines, we show that the unstable manifolds of the periodic points form a lamination outside \({\mathcal{O}}_{F}\) if and only if there are no heteroclinic tangencies.

     

Complicated behavior in cubic Hénon maps

Theoretical and Mathematical Physics - We study a generalized Hénon map in two-dimensional space. We find a region of the phase space where the nonwandering set exists, specify parameter...     

Controlling unpredictability in the randomly driven Hénon–Heiles system

Noisy scattering dynamics in the randomly driven Hénon–Heiles system is investigated in the range of initial energies where the motion is unbounded. In this paper we study, with the help of the exit basins and the escape time distributions, how an external perturbation, be it dissipation or periodic forcing with a random phase, can enhance or mitigate the unpredictability of a system that exhibit chaotic scattering. In fact, if basin boundaries have the Wada property, predictability becomes very complicated, since the basin boundaries start to intermingle, what means that there are points of different basins close to each other. The main responsible of this unpredictability is the external forcing with random phase, while the dissipation can recompose the basin boundaries and turn the system more predictable. Therefore, we do the necessary simulations to find out the values of dissipation and external forcing for which the exit basins present the Wada property. Through these numerical simulations, we show that the presence of the Wada basins have a specific relation with the damping, the forcing amplitude and the energy value. Our approach consists on investigating the dynamics of the system in order to gain knowledge able to control the unpredictability due to the Wada basins.     

Limit-cycle-like control for 2-dimensional discrete-time nonlinear control systems and its application to the Hénon map

In this paper, a control method that generates a desired limit-cycle-like behavior for a 2-dimensional discrete-time nonlinear control system is discussed. First, we define some notations and state the problem formulation. Next, a necessary and sufficient condition of existence of a control input that realizes a desired limit-cycle-like behavior is shown. We then derive a control algorithm to solve the problem on generating limit-cycle-like behaviors, and a modification of the algorithm is also shown. Finally, we apply the two types of algorithms to a chaotic system, the Hénon map, in order to indicate the availability of the proposed method. In addition, by using the control method, we also consider a stabilization problem for the Hénon map.     

A cubic Hénon-like map in the unfolding of degenerate homoclinic orbit with resonance

In this Note, we study the unfolding of a vector field that possesses a degenerate homoclinic (of inclination-flip type) to a hyperbolic equilibrium point where its linear part possesses a resonance. For the unperturbed system, the resonant term associated with the resonance vanishes. After suitable rescaling, the Poincaré return map is a cubic Hénon-like map. We deduce the existence of a strange attractor which persists in the Lebesgue measure sense. We also show the presence of an attractor with topological entropy close to $\log 3$. To cite this article: M. Martens et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Asymptotic behavior on the Hénon equation with supercritical exponent

The main purpose of this paper is to analyze the asymptotic behavior of the radial solution of Hénon equation −Δu = |x| ^α u ^p−1, u > 0, x ∈ B _R (0) ⊂ ℝ ⁿ (n ⩾ 3), u = 0, x ∈ ∂B _R (0), where $ p \to p(\alpha ) = \frac{{2(n + \alpha )}} {{n - 2}} $ p \to p(\alpha ) = \frac{{2(n + \alpha )}} {{n - 2}} from left side, α > 0.     

Three Positive Solutions of the One-Dimensional Generalized Hénon Equation

We study the one-dimensional generalized Hénon equation under the Dirichlet boundary condition. It is known that there exist at least three positive solutions if the coefficient function is even. In this paper, without the assumption of evenness, we prove the existence of at least three positive solutions.     

Recent Results on the Dynamics of Higher-dimensional Hénon Maps

We investigate different aspects of chaotic dynamics in Hénon maps of dimension higher than 2. First, we review recent results on the existence of homoclinic points in 2-d and 4-d such maps, by demonstrating how they can be located with great accuracy using the parametrization method. Then we turn our attention to perturbations of Hénon maps by an angle variable that are defined on the solid torus, and prove the existence of uniformly hyperbolic solenoid attractors for an open set of parameters.We thus argue that higher-dimensional Hénon maps exhibit a rich variety of chaotic behavior that deserves to be further studied in a systematic way.     

Hyperbolic invariant sets of the real generalized Hénon maps

In this paper, the conditions under which there exists a uniformly hyperbolic invariant set for the generalized Hénon map F(x, y) =  (y, ag(y) ? δx) are investigated, where g(y) is a monic real-coefficient polynomial of degree d ? 2, a and δ are non-zero parameters. It is proved that for certain parameter regions the map has a Smale horseshoe and a uniformly hyperbolic invariant set on which it is topologically conjugate to the two-sided fullshift on two symbols, where g(y) has at least two different non-negative or non-positive real zeros, and ∣a∣ is sufficiently large. Moreover, it is shown that if g(y) has only simple real zeros, then for sufficiently large ∣a∣, there exists a uniformly hyperbolic invariant set on which F is topologically conjugate to the two-sided fullshift on d symbols.     

Non-radial least energy solutions of the generalized Hénon equation

In this paper, we study the generalized Hénon equation with a radial coefficient function in the unit ball and show the existence of a positive non-radial solution. Our result is applicable to a wide class of coefficient functions. Our theorem ensures that if the ratio of the density of the coefficient function in $|x|<a$ to that in $a<|x|<1$ is small enough and a is sufficiently close to 1, then a least energy solution is not radially symmetric.     

Hénon映射的周期过程

本文研究了Ｈｅｎｏｎ映射Ｈ（ｘ，ｙ）＝（ａ＋ｂｙ－ｘ２，ｘ）的周期过程，给出了Ｈ有某些周期的充分必要条件或充分条件，得到其周期集的一段序结构．     

Chaotic Behavior of Hénon Mapping

There is an important criterion to check the chaotic behavior for a given discrete system defined by planar mapping. That is, the Smale-Birkhoff Theorem which says that a transversal homoclinic point implies chaos, see [5], [6]. Recently, [1] proved that a transversal N-cycle implies transversal homoclinic point.