Fractal basins of convergence of a seventh-order generalized Hénon-Heiles potential

分析广义Henon-Heiles势中五阶和七阶项的影响

Analyzing the Effects of Fifth and Seventh Order Terms in a Generalized Henon-Heiles Potential

本文研究了广义Henon-Heiles势中五阶和七阶项对动力学行为的影响，通过庞加莱截面和李雅普诺夫指数分析发现，五阶项促进规律性，而七阶项则增加混沌性。

On the dynamics of a seventh-order generalized Hénon-Heiles potential

This paper deals with the derivation and analysis of a seventh-order generalization of the Hénon-Heiles potential. The new potential has axial and reflection symmetries, and finite escape energy with three channels of escape. Based on SALI indicator and exits basins, the dynamic behavior of the seventh-order system is investigated qualitatively in cases of bounded and unbounded movement. Moreover, a quantitative analysis is carried out through the percentage of chaotic orbits and the basin entropy, respectively. After classifying large sets of initial conditions of orbits for several values of…

Basins of convergence of equilibrium points in the generalized Henon-Heiles system

We numerically explore the Newton-Raphson basins of convergence, related to the libration points (which act as attractors of the convergence process), in the generalized Hénon-Heiles system (GHH). The evolution of the position as well as of the linear stability of the equilibrium points is determined, as a function of the value of the perturbation parameter. The attracting regions, on the configuration $(x,y)$ plane, are revealed by using the multivariate version of the classical Newton-Raphson iterative algorithm. We perform a systematic investigation in an attempt to understand how the pertu…

Dynamical analysis of bounded and unbounded orbits in a generalized Hénon-Heiles system

The Hénon-Heiles potential was first proposed as a simplified version of the gravitational potential experimented by a star in the presence of a galactic center. Currently, this system is considered a paradigm in dynamical systems because despite its simplicity exhibits a very complex dynamical behavior. In the present paper, we perform a series expansion up to the fifth-order of a potential with axial and reflection symmetries, which after some transformations, leads to a generalized Hénon-Heiles potential. Such new system is analyzed qualitatively in both regimes of bounded and unbounded mot…

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 d…

描述引力波测地线的修正Henon-Heiles系统中的混沌

Chaos in a modified Henon-Heiles system describing geodesics in gravitational waves

研究了一个具有修正Henon-Heiles势的哈密顿系统。这描述了在真空引力PP波时空中自由测试粒子的运动，结构函数中有二次（“齐次”）和三次（“非齐次”）项。它示出，对于高于一定值的能量，运动是混沌的意义上说，分离盆地的边界可能逃脱成为分形。与标准的Henon-Heiles和猴子鞍系统的异同进行了讨论。还计算了流域边界的计盒维数。