Application of new dynamical spectra of orbits in Hamiltonian systems

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.Comment: Published in Nonlinear Dynamics (NODY) journal. arXiv admin note: text overlap with arXiv:1009.1993 by other author

Chaotic orbits in a 3D galactic dynamical model with a double nucleus

A 3D-dynamical model is constructed for the study of motion in the central regions of a disk galaxy with a double nucleus. Using the results of the 2D-model, we find the regions of initial conditions in the (x,px,z,py)=EJ, (y=pz=0) phase space producing regular or chaotic orbits. The majority of stars are on chaotic orbits. All chaotic orbits come arbitrary close to one or to both nuclei. Regular orbits are those starting near the stable periodic orbits of the 2D-system with small values of z0. All regular orbits circulate around only one of the two nuclei.Comment: Published in Mechanics Research Communications journa

Investigating the nature of motion in 3D perturbed elliptic oscillators displaying exact periodic orbits

We study the nature of motion in a 3D potential composed of perturbed elliptic oscillators. Our technique is to use the results obtained from the 2D potential in order to find the initial conditions generating regular or chaotic orbits in the 3D potential. Both 2D and 3D potentials display exact periodic orbits together with extended chaotic regions. Numerical experiments suggest, that the degree of chaos increases rapidly, as the energy of the test particle increases. About 97% of the phase plane of the 2D system is covered by chaotic orbits for large energies. The regular or chaotic character of the 2D orbits is checked using the S(c) dynamical spectrum, while for the 3D potential we use the S(c) spectrum, along with the P(f) spectral method. Comparison with other dynamical indicators shows that the S(c) spectrum gives fast and reliable information about the character of motion.Comment: Published in Nonlinear Dynamics (NODY) journa

The intersection surfaces in a 4-dimensional homoclinic/heteroclinic tangle

We study the homoclinic/heteroclinic tangle in a three degrees of freedom system corresponding to a 4-dimensional Poincaré map. In particular, we look at the 2-dimensional homoclinic/heteroclinic intersection surfaces between the stable and unstable manifolds of the most important codimension-2 invariant sets and study how the internal structures of these invariant sets are transported into the intersection surfaces. This provides a pictorial overview of the 2-dimensional continuum of homoclinic/heteroclinic connections via this particular intersection surface. As an example of demonstration, we use a model for a barred galaxy containing a three degrees of freedom version of a ternary symmetric horseshoe. © 2022, The Author(s), under exclusive licence to Springer Nature B.V

Applying chaos indicators to Bianchi cosmological models

Using standard chaos indicators, we perform a numerical investigation of the Bianchi IX Universe in the limit where one of the structure constants tends to zero. In this limit, we recover Bianchi VII0, which can be obtained from a Lie algebra contraction. We find that in this limit, the chaoticity of the system is preserved, but the basin structures of the grid classification are changed. © 2022 Elsevier Lt

Manifold dynamics and periodic orbits in a multiwell potential

In this article, we explore the dynamics as well as the geometry of the invariant manifolds that determine the escapes from a multiwell potential. We also present the network of both symmetric and asymmetric solutions of the system, while at the same time we extract valuable information about the periodic solutions, such as their locations, multiplicity, and linear stability. © 2022 Elsevier Lt