Application of new dynamical spectra of orbits in Hamiltonian systems

TL;DR

This paper introduces two new dynamical spectra, S(g) and S(w), for analyzing orbit properties in Hamiltonian systems with two and three degrees of freedom, demonstrating their efficiency in distinguishing orbit types.

Contribution

The paper presents novel dynamical spectra, S(g) and S(w), specifically designed for Hamiltonian systems, improving orbit classification accuracy and computational speed over existing methods.

Findings

Both spectra reliably distinguish between regular and chaotic orbits.

The spectra detect tiny ordered regions within chaotic seas.

They require very short computation times.

Abstract

In the present article, we investigate the properties of motion in Hamiltonian systems of two and three degrees of freedom, using the distribution of the values of two new dynamical parameters. The distribution functions of the new parameters, define the S(g) and the S(w) dynamical spectra. The first spectrum definition, that is the S(g) spectrum, will be applied in a Hamiltonian system of two degrees of freedom (2D), while the S(w) dynamical spectrum will be deployed in a Hamiltonian system of three degrees of freedom (3D). Both Hamiltonian systems, describe a very interesting dynamical system which displays a large variety of resonant orbits, different chaotic components and also several sticky regions. We test and prove the efficiency and the reliability of these new dynamical spectra, in detecting tiny ordered domains embedded in the chaotic sea, corresponding to complicated resonant orbits of higher multiplicity. The results of our extensive numerical calculations, suggest that both dynamical spectra are fast and reliable discriminants between different types of orbits in Hamiltonian systems, while requiring very short computation time in order to provide solid and conclusive evidence regarding the nature of an orbit. Furthermore, we establish numerical criteria in order to quantify the results obtained from our new dynamical spectra. A comparison to other previously used dynamical indicators, reveals the leading role of the new spectra.