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.     

Multistability annihilation in the Hénon map through parameters modulation

We consider a situation in which the two parameters of a Hénon map are modulated by the output of another Hénon map. Two cases are considered. Firstly, we investigate the behavior of the Hénon map when its parameters are modulated by another Hénon map, this last working in a high dissipative regime. Secondly, we use a Hénon map working in a low dissipative regime as the modulation. We show that, regardless of the considered case, multistability can be suppressed by the modulation.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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．     

Multi-peak solutions for the Hénon equation with slightly subcritical growth

We study the Dirichlet problem for the Hénon equation

非线性Hénon方程解的Liouville定理

该文研究了二阶和四阶非线性Henon-Lane-Emden方程有限Morse指标解的Liouville定理.利用一种新方法,即使用单调公式、Pohozaev恒等式和doubling引理等相结合证明了其结果.     

Least energy solutions of the generalized Hénon equation in reflectionally symmetric or point symmetric domains

We study the generalized Hénon equation in reflectionally symmetric or point symmetric domains and prove that a least energy solution is neither reflectionally symmetric nor even. Moreover, we prove the existence of a positive solution with prescribed exact symmetry.     

Non radial positive solutions for the Hénon equation with critical growth

We study the Dirichlet problem in a ball for the Hénon equation with critical growth and we establish, under some conditions, the existence of a positive, non radial solution. The solution is obtained as a minimizer of the quotient functional associated to the problem restricted to appropriate subspaces of H₀¹ invariant for the action of a subgroup of . Analysis of compactness properties of minimizing sequences and careful level estimates are the main ingredients of the proof. Received: 18 October 2003, Accepted: 5 July 2004, Published online: 3 September 2004 Mathematics Subject Classification (2000): 35J20, 35B33 This research was supported by MIUR, Project "Variational Methods and Nonlinear Differential Equations".