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

Abstract

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 motion via the Poincaré sections method and plotting the exit basins. On the other hand, the quantitative analysis is performed through the Lyapunov exponents and the basin entropy, respectively. We find that in both regimes the chaoticity of the system decreases as long as the test particle energy gets far from the critical energy. Additionally, we may conclude that despite the inclusion of higher order terms in the series expansion, the new system shows wider zones of regularity (islands) than the ones present in the Hénon-Heiles system.

Exit basins for the Sitnikov problem with variable mass

“… Our findings are in agreement with previous studies related to the influence of perturbing terms in the dynamics of open and closed astrophysical systems [17,18,19].…”

Fractal Basins of Convergence of a Seventh-Order Generalized Hénon–Heiles Potential

“…Around 1980, Verhulst [10] expanded the potential (2) up to the fourth order seeking to study resonances 1 : 1, 1 : 2, 1 : 3, and 2 : 1. Some years ago, a generalized Hénon-Heiles potential was derived by expanding the effective potential up to the fifth order, aiming to study the equilibrium points and basins of convergence of the new potential [11] and to analyze the dynamical effect on bounded and unbounded orbits of including higher-order terms in the series expansion [12]. More recently, a seventhorder version of the stationary axisymmetric potential was presented [13], and it was found that when higher-order contributions of the potential are taken into account, the chaoticity of the system is reduced in comparison with the lower-order version of the Hénon-Heiles system.…”

Basins of convergence of equilibrium points in the generalized Hénon–Heiles system

“…Potential (2) is derived as a Taylor expansion up to the 5-th order of a general potential with axial and reflection symmetries. More information about the exact expansion and the derivation of the generalized potential is given in [11].…”

“…The Hénon-Heiles Hamiltonian is a two-dimensional time-independent dynamical system, originally proposed as a simplified version of the gravitational potential experimented by a star orbiting around an axially symmetric galaxy (see e.g., [13]). An extension of this potential up to the fourth-order was performed by Verhulst [19], while in [11] we generalized the Hénon-Heiles potential up to the fifth-order. In the present paper, we examine the basins of convergence in the fifth-order generalization of the Hénon-Heiles Hamiltonian (in all that follows GHH).…”