arXiv subjects

Matthaios Katsanikas

Publications and source records attributed to Matthaios Katsanikas.

14 recordsLinked to original sources

arXiv2024

The evolution of the phase space structure along pitchfork and period-doubling bifurcations in a 3D galactic bar potential

We investigate how the phase space structure of a 3D autonomous Hamiltonian system evolves across a series of successive 2D and 3D pitchfork and period-doubling bifurcations, as the transition of the parent families of periodic orbits (POs) from stability to simple instability leads to the creation of new stable POs. Our research illustrates the consecutive alterations in the phase space structure near POs as the stability of the main family of POs changes. This process gives rise to new families of POs within the system, either maintaining the same or exhibiting higher multiplicity compared to their parent families. Tracking such a phase space transformation is challenging in a 3D system. By utilizing the color and rotation technique to visualize the 4D Poincar'e surfaces of section of the system, i.e. projecting them onto a 3D subspace and employing color to represent the fourth dimension, we can identify distinct structural patterns. Perturbations of parent and bifurcating stable POs result in the creation of tori characterized by a smooth color variation on their surface. Furthermore, perturbations of simple unstable parent POs beyond the bifurcation point which lead to the birth of new stable families of POs, result in the formation of figure-8 structures of smooth color variations. These figure-8 formations surround well-shaped tori around the bifurcated stable POs, losing their well-defined forms for energies further away from the bifurcation point. We also observe that even slight perturbations of highly unstable POs create a cloud of mixed color points, which rapidly move away from the location of the PO. Our study introduces, for the first time, a systematic visualization of 4D surfaces of section within the vicinity of higher multiplicity POs. It elucidates how, in these cases, the coexistence of regular and chaotic orbits contributes to shaping the phase space landscape.

nlin.CD↗

arXiv2024

Bifurcation of Dividing Surfaces Constructed from Period-Doubling Bifurcations of Periodic Orbits in a Caldera Potential Energy Surface

In this work we analyze the bifurcation of dividing surfaces that occurs as a result of two period-doubling bifurcations in a 2D caldera-type potential. We study the structure, the range, the minimum and maximum extents of the periodic orbit dividing surfaces before and after a subcritical period-doubling bifurcation of the family of the central minimum of the potential energy surface. Furthermore, we repeat the same study for the case of a supercritical perioddoubling bifurcation of the family of the central minimum of the potential energy surface. We will discuss and compare the results for the two cases of bifurcations of dividing surfaces.

nlin.CD↗

arXiv2023

Performance of chaos diagnostics based on Lagrangian descriptors. Application to the 4D standard map

We investigate the ability of simple diagnostics based on Lagrangian descriptor (LD) computations of initially nearby orbits to detect chaos in conservative dynamical systems with phase space dimensionality higher than two. In particular, we consider the recently introduced methods of the difference ($D_L^n$) and the ratio ($R_L^n$) of the LDs of neighboring orbits, as well as a quantity ($S_L^n$) related to the finite-difference second spatial derivative of the LDs, and use them to determine the chaotic or regular nature of ensembles of orbits of a prototypical area-preserving map model, the 4-dimensional (4D) symplectic standard map. Using the distributions of the indices' values we determine appropriate thresholds to discriminate between regular and chaotic orbits, and compare the obtained characterization against that achieved by the Smaller Alignment Index (SALI) method of chaos detection, by recording the percentage agreement $P_A$ between the two classifications. We study the influence of various factors on the performance of these indices, and show that the increase of the final number of orbit iterations T and the order n of the indices (i.e. the dimensionality of the space where the considered nearby orbits lie), as well as the decrease of the distance $σ$ of neighboring orbits, increase the $P_A$ values along with the required computational effort. Balancing between these two factors we find appropriate T, n and $σ$ values, which allow the efficient use of the $D_L^n$, $R_L^n$ and $S_L^n$ indices as short time and computationally cheap chaos diagnostics achieving $P_A \gtrsim 90 \%$, with $D_L^n$ and $S_L^n$ having larger $P_A$ values than $R_L^n$. Our results show that the three LDs-based indices perform better for systems with large percentages of chaotic orbits.

astro-ph.EP↗

arXiv2023

Navigating Phase Space Transport with the Origin-Fate Map

We introduce and demonstrate the usage of the origin-fate map (OFM) as a tool for the detailed investigation of phase space transport in reactant-product type systems. For these systems, which exhibit clearly defined start and end states, it is possible to build a comprehensive picture of the lobe dynamics by considering backward and forward integration of sets of initial conditions to index their origin and fate. We illustrate the method and its utility in the study of a two degrees of freedom caldera potential with four exits, demonstrating that the OFM not only recapitulates results from classical manifold theory, but even provides more detailed information about complex lobe structures. The OFM allows the detection of dynamically significant transitions caused by the creation of new lobes, and is also able to guide the prediction of the position of unstable periodic orbits (UPOs). Further, we compute the OFM on the periodic orbit dividing surface (PODS) associated with the transition state of a caldera entrance, which allows for a powerful analysis of reactive trajectories. The intersection of the manifolds corresponding to this UPO with other manifolds in the phase space results in the appearance of lobes on the PODS, which are directly classified by the OFM. This allows computations of branching ratios and the exploration of a fractal cascade of lobes as the caldera is stretched, which results in fluctuations in the branching ratio and chaotic selectivity. The OFM is found to be a simple and very useful tool with a vast range of descriptive and quantitative applications.

math.DS↗

arXiv2022

Quantifying chaos using Lagrangian descriptors

We present and validate simple and efficient methods to estimate the chaoticity of orbits in low dimensional dynamical systems from computations of Lagrangian descriptors (LDs) on short time scales. Two quantities are proposed for determining the chaotic or regular nature of orbits in a system's phase space, which are based on the values of the LDs of these orbits and of nearby ones: The difference (DNLD) and ratio (RNLD) of neighboring orbits' LDs. We find that, typically, these indicators are able to correctly characterize the chaotic or regular nature of orbits to better than 90 % agreement with results obtained by implementing the Smaller Alignment Index (SALI) method, which is a well established chaos detection technique. Further investigating the performance of the two introduced quantities we discuss the effects of the total integration time and of the spacing between the used neighboring orbits on the accuracy of the methods, finding that even typical short time, coarse grid LD computations are sufficient to provide a reliable quantification of a system's chaotic component, using less CPU time than the SALI. In addition to quantifying chaos, the introduced indicators have the ability to reveal details about the systems' local and global chaotic phase space structure. Our findings clearly suggest that LDs can also be used to quantify and investigate chaos in continuous and discrete low dimensional dynamical systems.

nlin.CD↗

arXiv2022

The Nature of Reactive and Non-reactive Trajectories for a Three Dimensional Caldera Potential Energy Surface

We used for the first time the method of periodic orbit dividing surfaces in a non-integrable Hamiltonian system with three degrees of freedom. We have studied the structure of these four dimensional objects in the five dimensional phase space. This method enabled us to detect the reactive and non-reactive trajectories in a three dimensional Caldera potential energy surface. We distinguished four distinct types of trajectory behavior. Two of the types of trajectories could only occur in a three dimensional Caldera potential energy surface and not in a two dimensional surface, and we have shown that this is a result of homoclinic intersections. These homoclinic intersections were analyzed with the method of Lagrangian descriptors. Finally, we were able to detect and describe the phenomenon of dynamical matching in the three dimensional Caldera potential energy surface, which is an important mechanism for understanding the reaction dynamics of organic molecules.

nlin.CD↗

arXiv2022

The Influence of a Parameter that Controls the Asymmetry of a Potential Energy Surface with an Entrance Channel and Two Potential Wells

In this paper we study an asymmetric valley-ridge inflection point (VRI) potential, whose energy surface (PES) features two sequential index-1 saddles (the upper and the lower), with one saddle having higher energy than the other and two potential wells separated by the lower index-1 saddle. We show how the depth and the flatness of our potential changes as we modify the parameter that controls the asymmetry as well as how the branching ratio (ratio of the trajectories that enter each well) is changing as we modify the same parameter and its correlation with the area of the lobes as they have formulated by the stable and unstable manifolds that have been extracted from the gradient of the LD scalar fields.

math.DS↗

arXiv2021

The time evolution of the trajectories after the selectivity in a symmetric potential energy surface with a post-transition-state bifurcation

Selectivity is an important phenomenon in chemical reaction dynamics. This can be quantified by the branching ratio of the trajectories that visit one or the other wells to the total number of trajectories in a system with a potential with two sequential index-1 saddles and two wells (top well and bottom well). In our case, the branching ratio is 1:1 because of the symmetry of our potential energy surface. The mechanisms of transport and the behavior of the trajectories in this kind of systems have been studied recently. In this paper we study the time evolution after the selectivity as energy varies using periodic orbit dividing surfaces. We investigate what happens after the first visit of a trajectory to the region of the top or the bottom well for different values of energy. We answer the natural question, what is the destiny of these trajectories?

nlin.CD↗

arXiv2021

Bifurcation of dividing surfaces constructed from a pitchfork bifurcation of periodic orbits in a symmetric potential energy surface with a post-transition-state bifurcation

In this work we analyze the bifurcation of dividing surfaces that occurs as a result of a pitchfork bifurcation of periodic orbits in a two degrees of freedom Hamiltonian System. The potential energy surface of the system that we consider has four critical points: two minima, a high energy saddle and a lower energy saddle separating two wells (minima). In this paper we study the structure, the range, and the minimum and maximum extent of the periodic orbit dividing surfaces of the family of periodic orbits of the lower saddle as a function of the total energy.

nlin.CD↗

arXiv2021

Revealing the phase space structure of Hamiltonian systems using the action

In this work, we analyse the properties of the Maupertuis' action as a tool to reveal the phase space structure for Hamiltonian systems. We construct a scalar field with the action's values along the trajectories in the phase space. The different behaviour of the trajectories around important phase space objects like unstable periodic orbits, their stable and unstable manifolds, and KAM islands generate characteristic patterns on the scalar field constructed with the action. Using these different patterns is possible to identify the skeleton of the phase space and understand the dynamics. Also, we present a simple argument based on the conservation of the energy and the behaviour of the trajectories to understand the values of their actions. In order to show how this tool reveals the phase space structures and its effectiveness, we compare the scalar field constructed with the actions with Poincare maps for the same set of initial conditions in the phase space of an open Hamiltonian system with 2 degrees of freedom.

nlin.CD↗

arXiv2018

Phase Space Analysis of the Non-Existence of Dynamical Matching in a Stretched Caldera Potential Energy Surface

In this paper we continue our studies of the two dimensional caldera potential energy surface in a parametrized family that allows for a study of the effect of symmetry on the phase space structures that govern how trajectories enter, cross, and exit the region of the caldera. As a particular form of trajectory crossing, we are able to determine the effect of symmetry and phase space structure on dynamical matching. We show that there is a critical value of the symmetry parameter which controls the phase space structures responsible for the manner of crossing, interacting with the central region (including trapping in this region) and exiting the caldera. We provide an explanation for the existence of this critical value in terms of the behavior of the Henon stability parameter for the associated periodic orbits.

nlin.CD↗

arXiv2018

Phase Space Structure and Transport in a Caldera Potential Energy Surface

We study phase space transport in a 2D caldera potential energy surface (PES) using techniques from nonlinear dynamics. The caldera PES is characterized by a flat region or shallow minimum at its center surrounded by potential walls and multiple symmetry related index one saddle points that allow entrance and exit from this intermediate region.We have discovered four qualitatively distinct cases of the structure of the phase space that govern phase space transport. These cases are categorized according to the total energy and the stability of the periodic orbits associated with the family of the central minimum, the bifurcations of the same family, and the energetic accessibility of the index one saddles. In each case we have computed the invariant manifolds of the unstable periodic orbits of the central region of the potential and the invariant manifolds of the unstable periodic orbits of the families of periodic orbits associated with the index one saddles. We have found that there are three distinct mechanisms determined by the invariant manifold structure of the unstable periodic orbits govern the phase space transport. The first mechanism explains the nature of the entrance of the trajectories from the region of the low energy saddles into the caldera and how they may become trapped in the central region of the potential. The second mechanism describes the trapping of the trajectories that begin from the central region of the caldera, their transport to the regions of the saddles, and the nature of their exit from the caldera. The third mechanism describes the phase space geometry responsible for the dynamical matching of trajectories originally proposed by Carpenter and described in Collins et al. (2014) for the two dimensional caldera PES that we consider.

nlin.CD↗

arXiv2016

The dynamics of a spinning particle in a linear in spin Hamiltonian approximation

We investigate for order and chaos the dynamical system of a spinning test particle of mass $m$ moving in the spacetime background of a Kerr black hole of mass M. This system is approximated in our investigation by the linear in spin Hamiltonian function provided in [E. Barausse, and A. Buonanno, Phys.Rev. D 81, 084024 (2010)]. We study the corresponding phase space by using 2D projections on a surface of section and the method of color and rotation on a 4D Poincaré section. Various topological structures coming from the non-integrability of the linear in spin Hamiltonian are found and discussed. Moreover, an interesting result is that from the value of the dimensionless spin $S/(m M)=10^{-4}$ of the particle and below, the impact of the non-integrability of the system on the motion of the particle seems to be negligible.

gr-qc↗

arXiv2015

The Cosmic Battery in Astrophysical Accretion Disks

The aberrated radiation pressure at the inner edge of the accretion disk around an astrophysical black hole imparts a relative azimuthal velocity on the electrons with respect to the ions which gives rise to a ring electric current that generates large scale poloidal magnetic field loops. This is the Cosmic Battery established by Contopoulos and Kazanas in 1998. In the present work we perform realistic numerical simulations of this important astrophysical mechanism in advection-dominated accretion flows-ADAF. We confirm the original prediction that the inner parts of the loops are continuously advected toward the central black hole and contribute to the growth of the large scale magnetic field, whereas the outer parts of the loops are continuously diffusing outward through the turbulent accretion flow. This process of inward advection of the axial field and outward diffusion of the return field proceeds all the way to equipartition, thus generating astrophysically significant magnetic fields on astrophysically relevant timescales. We confirm that there exists a critical value of the magnetic Prandtl number between unity and 10 in the outer disk above which the Cosmic Battery mechanism is suppressed.

astro-ph.HE↗