Classifying orbits in the classical Hénon–Heiles Hamiltonian system

Abstract

The Hénon-Heiles potential is undoubtedly one of the most simple, classical and characteristic Hamiltonian systems. The aim of this work is to reveal the influence of the value of the total orbital energy, which is the only parameter of the system, on the different families of orbits, by monitoring how the percentage of chaotic orbits, as well as the percentages of orbits composing the main regular families evolve when energy varies. In particular, we conduct a thorough numerical investigation distinguishing between ordered and chaotic orbits, considering only bounded motion for several energy levels. The smaller alignment index (SALI) was computed by numerically integrating the equations of motion as well as the variational equations to extensive samples of orbits in order to distinguish safely between ordered and chaotic motion. In addition, a method based on the concept of spectral dynamics that utilizes the Fourier transform of the time series of each coordinate is used to identify the various families of regular orbits and also to recognize the secondary resonances that bifurcate from them. Our exploration takes place both in the physical (x, y) and the phase (y,ẏ) space for a better understanding of the orbital properties of the system. It was found, that for low energy levels the motion is entirely regular being the box orbits the most populated family, while as the value of the energy increases chaos and several resonant families appear. We also observed, that the vast majority of the resonant orbits belong in fact in bifurcated families of the main 1:1 resonant family. We have also compared our results with previous similar outcomes obtained using different chaos indicators.

Presence of horizon makes particle motion chaotic

“…This is consistent with the KAM theory which states that the nonlinear perturbation in the integrable dynamical system results to the generation of the chaos [27]. There are several examples reported earlier in support of this claim; like addition of non-linear perturbation in Henon-Heiles potential leads to chaotic behaviour of the system at high energy [29], appearance of chaos in double pendulum (Hamiltonian) for large oscillation [30] (also see [31] for a discussion in this direction). In our case, for small perturbation, i.e., when the harmonic term is small we observe a regular tori.…”

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

“…Nowadays it can be considered one of the most cited works in the field of complex systems, where a huge amount of research has been devoted to discriminate between regular and chaotic motion or to study the escape dynamics of orbits, see e.g. [5][6][7][8][9][10][11][12]. Although its application was first oriented to the field of galactic dynamics, its applications include semiclassical and quantum mechanics [13][14][15].…”

Phase Portraits of the Henon-Heiles Potential

“…This potential is analytically simple, so that the orbits can be calculated quite easily, but it is still quite complex, so that the types of orbits are nontrivial. This potential is now known as the potential of Henon and Heiles [1][2][3].…”

“…It is seen that 0 h  the Hamiltonian is symmetric with respect to x x   , and also exhibits a symmetry of rotation at 2 / 3  . Below are the dependencies of the coordinates of the functions in time for the systems of equations (2). To study the Henon-Heiles system, the Poincaré section method is used.…”

“…In this case, the phase trajectories are calculated by numerical methods. To solve the systems of equations (2), boundary conditions are chosen so that they satisfy equation (3). Further, the systems of equation 2are solved on the basis of the Runge-Kutta method.…”