arXiv subjects

M. Katsanikas

Publications and source records attributed to M. Katsanikas.

At least 19 recordsLinked to original sources

arXiv2022

The Phase Space Structure in the vicinity of vertical Lyapunov orbits around L1,2 in a barred galaxy model

We study the phase space structure and the orbital diffusion from the vicinity of the vertical Lyapunov periodic orbits around the unstable Lagrangian points L1,2 in a 3D barred galaxy model. By perturbing the initial conditions of these periodic orbits, we detected the following five types of orbital structures in the 4D spaces of section: (i) Ring-like structures, sticky for large time intervals to the unstable invariant manifolds of the simple and double unstable vertical Lyapunov periodic orbits. (ii) 2D tori belonging to quasi-periodic orbits around stable periodic orbits existing in the region. They are associated either with vertical stable periodic orbits around L4,5 or with "stable anomalous" periodic orbits. (iii) Orbits sticky for large time intervals to these tori, forming "sticky tori", before they slowly depart from them. (iv) Clouds of points that have a strong chaotic behavior. Such clouds of consequents have slow diffusion speeds, because they are hindered by the presence of the tori around the "stable anomalous" periodic orbits. (v) Toroidal zones consisting of points that stick for long time on the unstable invariant manifolds of the "unstable anomalous" periodic orbits. By continuing the integration, we find that eventually they become strongly chaotic, retaining however small diffusion speeds, due to the presence of the tori around the stable anomalous periodic orbits.

astro-ph.GA↗

arXiv2022

Phase space transport in a symmetric Caldera potential with three index-1 saddles and no minima

We apply the method of Lagrangian Descriptors (LDs) to a symmetric Caldera-type potential energy surface which has three index-1 saddles surrounding a relatively flat region that contains no minimum. Using this method we show the phase space transport mechanism that is responsible for the existence and non-existence of the phenomenon of dynamical matching for this form of Caldera potential energy surface.

nlin.CD↗

arXiv2021

The Generalization of the Periodic Orbit Dividing Surface in Hamiltonian Systems with three or more degrees of freedom -- I

We present a method that generalizes the periodic orbit dividing surface construction for Hamiltonian systems with three or more degrees of freedom. We construct a torus using as a basis a periodic orbit and we extend this to a $2n-2$ dimensional object in the $2n-1$ dimensional energy surface. We present our methods using benchmark examples for two and three degree of freedom Hamiltonian systems to illustrate the corresponding algorithm for this construction. Towards this end we use the normal form quadratic Hamiltonian system with two and three degrees of freedom. We found that the periodic orbit dividing surface can provide us the same dynamical information as the dividing surface constructed using normally hyperbolic invariant manifolds. This is significant because, in general, computations of normally hyperbolic invariant manifolds are very difficult in Hamiltonian systems with three or more degrees of freedom. However, our method avoids this computation and the only information that we need is the location of one periodic orbit.

nlin.CD↗

arXiv2021

Bifurcation Study on a Degenerate Double van der Waals Cirque Potential Energy Surface using Lagrangian Descriptors

In this paper, we explore the dynamics of a Hamiltonian system after a double van der Waals potential energy surface degenerates into a single well. The energy of the system is increased from the bottom of the potential well up to the dissociation energy, which occurs when the system becomes open. In particular, we study the bifurcations of the basic families of periodic orbits of this system as the energy increases using Lagrangian descriptors and Poincaré maps. We investigate the capability of Lagrangian descriptors to find periodic orbits of bifurcating families for the case of resonant, saddle-node and pitchfork bifurcations.

nlin.CD↗

arXiv2021

The bifurcations of the critical points and the role of the depth in a symmetric Caldera potential energy surface

In this work, we continue the study of the bifurcations of the critical points in a symmetric Caldera potential energy surface. In particular, we study the influence of the depth of the potential on the trajectory behavior before and after the bifurcations of the critical points. We observe two different types of trajectory behavior: dynamical matching and the non-existence of dynamical matching. Dynamical matching is a phenomenon that limits the way in which a trajectory can exit the Caldera based solely on how it enters the Caldera. Furthermore, we discuss two different types of symmetric Caldera potential energy surface and the transition from the one type to the other through the bifurcations of the critical points.

nlin.CD↗

arXiv2021

Visualizing the Phase Space of the HeI_2 van der Waals Complex using Lagrangian Descriptors

In this paper we demonstrate the capability of the method of Lagrangian descriptors to unveil the phase space structures that characterize transport in high-dimensional symplectic maps. In order to illustrate its use, we apply it to a four-dimensional symplectic map model that is used in chemistry to explore the nonlinear dynamics of van der Waals complexes. The advantage of this technique is that it allows us to easily and effectively extract the invariant manifolds that determine the dynamics of the system under study by means of examining the intersections of the underlying phase space structures with low-dimensional slices. With this approach, one can perform a full computational phase space tomography from which three-dimensional representations of the higher-dimensional phase space can be systematically reconstructed. This analysis may be of much help for the visualization and understanding of the nonlinear dynamical mechanisms that take place in high-dimensional systems. In this context, we demonstrate how this tool can be used to detect whether the stable and unstable manifolds of the system intersect forming turnstile lobes that enclose a certain phase space volume, and the nature of their intersection.

nlin.CD↗

arXiv2021

From Poincare Maps to Lagrangian Descriptors: The Case of the Valley Ridge Inflection Point Potential

In this paper we compare the method of Lagrangian descriptors with the classical method of Poincare maps for revealing the phase space structure of two degree-of-freedom Hamiltonian systems. The comparison is carried out by considering the dynamics of a two degree-of-freedom system having a valley ridge inflection point (VRI) potential energy surface. VRI potential energy surfaces have four critical points: a high energy saddle and a lower energy saddle separating two wells. In between the two saddle points is a valley ridge inflection point that is the point where the potential energy surface geometry changes from a valley to a ridge. The region between the two saddles forms a reaction channel and the dynamical issue of interest is how trajectories cross the high energy saddle, evolve towards the lower energy saddle, and select a particular well to enter. Lagrangian descriptors and Poincare maps are compared for their ability to determine the phase space structures that govern this dynamical process.

nlin.CD↗

arXiv2021

The Influence of a Pitchfork Bifurcation of the Critical Points of a Symmetric Caldera Potential Energy Surface on Dynamical Matching

Many organic chemical reactions are governed by potential energy surfaces that have a region with the topographical features of a caldera. If the caldera has a symmetry then trajectories transiting the caldera region are observed to exhibit a phenomenon that is referred to as dynamical matching. Dynamical matching is a constraint that restricts the way in which a trajectory can exit the caldera based solely on how it enters the caldera. In this paper we show that bifurcations of the critical points of the caldera potential energy surface can destroy dynamical matching even when the symmetry of the caldera is not affected by the bifurcation.

nlin.CD↗

arXiv2020

Detection of Dynamical Matching in a Caldera Hamiltonian System using Lagrangian Descriptors

The goal of this paper is to apply the method of Lagrangian descriptors to reveal the phase space mechanism by which a Caldera-type potential energy surface (PES) exhibits the dynamical matching phenomenon. Using this technique, we can easily establish that the non-existence of dynamical matching is a consequence of heteroclinic connections between the unstable manifolds of the unstable periodic orbits (UPOs) of the upper index-1 saddles (entrance channels to the Caldera) and the stable manifolds of the family of UPOs of the central minimum of the Caldera, resulting in the temporary trapping of trajectories. Moreover, dynamical matching will occur when there is no heteroclinic connection, which allows trajectories to enter and exit the Caldera without interacting with the shallow region of the central minimum. Knowledge of this phase space mechanism is relevant because it allows us to effectively predict the existence, and non-existence, of dynamical matching. In this work we explore a stretched Caldera potential by means of Lagrangian descriptors, allowing us to accurately compute the critical value for the stretching parameter for which dynamical matching behavior occurs in the system. This approach is shown to provide a tremendous advantage for exploring this mechanism in comparison to other methods from nonlinear dynamics that use phase space dividing surfaces.

nlin.CD↗

arXiv2020

Phase Space Analysis of the Dynamics on a Potential Energy Surface with an Entrance Channel and Two Potential Wells

In this paper we unveil the geometrical template of phase space structures that governs transport in a Hamiltonian system described by a potential energy surface with an entrance/exit channel and two wells separated by an index-1 saddle. For the analysis of the nonlinear dynamics mechanisms, we apply the method of Lagrangian descriptors, a trajectory-based scalar diagnostic tool that is capable of providing a detailed phase space tomography of the interplay between the invariant manifolds of the system. Our analysis reveals that, the stable and unstable manifolds of two families of unstable periodic orbits (UPOs) that exist in the regions of the wells are responsible for controlling the access to the wells of trajectories that enter the system through the channel. In fact, we demonstrate that the heteroclinic and homoclinic connections that arise in the system between the manifolds of the families of UPOs characterize the branching ratio, a relevant quantity used to measure product distributions in chemical reaction dynamics.

nlin.CD↗

arXiv2020

Tuning the Branching Ratio in a Symmetric Potential Energy Surface with a Post-Transition State Bifurcation using External Time Dependence

Chemical selectivity, as quantified by a branching ratio, is a phenomenon relevant for many organic chemical reactions. It may be exhibited on a potential energy surface that features a valley-ridge inflection point in the region between two sequential index-1 saddles, with one saddle having higher energy than the other. Reaction occurs when a trajectory crosses the region of the higher energy saddle (the entrance channel) and approaches the lower energy saddle. On both sides of the lower energy saddle, there are two wells and the question we address in this work is that, given an initial ensemble of trajectories, what is the relative fraction of trajectories that enter each well. For a symmetric PES this fraction is 1:1. We consider a symmetric PES subjected to a time-periodic forcing characterized by an amplitude, frequency, and phase. In this letter we analyse how the branching ratio depends on these three parameters.

physics.chem-ph↗

arXiv2020

The Phase Space Mechanism for Selectivity in a Symmetric Potential Energy Surface with a Post-Transition-State Bifurcation

Chemical selectivity is a phenomenon displayed by potential energy surfaces (PES) that is relevant for many organic chemical reactions whose PES feature a valley-ridge inflection point (VRI) in the region between two sequential index-1 saddles. In this letter we describe the underlying dynamical phase space mechanism that qualitatively determines the product distributions resulting from bifurcating reaction pathways. We show that selectivity is a consequence of the heteroclinic and homoclinic connections established between the invariant manifolds of the families of unstable periodic orbits (UPOs) present in the system. The geometry of the homoclinic and heteroclininc connections is determined using the technique of Lagrangian descriptors, a trajectory-based scalar technique with the capability of unveiling the geometrical template of phase space structures that characterizes transport.

physics.chem-ph↗

arXiv2019

The Dynamical Matching Mechanism in Phase Space for Caldera-Type Potential Energy Surfaces

Dynamical matching occurs in a variety of important organic chemical reactions. It is observed to be a result of a potential energy surface (PES) having specific geometric features. In particular, a region of relative flatness where entrance and exit to this region is controlled by index-one saddles. Examples of potential energy surfaces having these features are the so-called caldera potential energy surfaces. We develop a predictive level of understanding of the phenomenon of dynamical matching in a caldera potential energy surface. We show that the phase space structure that governs dynamical matching is a particular type of heteroclinic trajectory which gives rise to trapping of trajectories in the central region of the caldera PES. When the heteroclinic trajectory is broken, as a result of parameter variations, then dynamical matching occurs.

nlin.CD↗

arXiv2014

The phase-space of boxy-peanut and X-shaped bulges in galaxies II. The relation between face-on and edge-on boxiness

We study the dynamical mechanisms that reinforce the formation of boxy structures in the \textit{inner} regions, roughly in the middle, of bars observed nearly \textit{face-on}. Outer boxiness, at the ends of the bars, is usually associated with orbits at the inner, radial 4:1 resonance region and can be studied with 2D dynamics. However, in the middle of the bar dominate 3D orbits that give boxy/peanut bulges in the edge-on views of the models. In the present paper we show that 3D quasi-periodic, as well as 3D chaotic orbits sticky to the x1v1 and x1v1$^{\prime}$ tori, especially from the Inner Lindblad Resonance (ILR) region, have boxy projections on the equatorial plane of the bar. The majority of vertically perturbed 2D orbits, initially on the equatorial plane in the ILR resonance region, enhance boxy features in face-on bars. Orbits that build a bar by supporting sharp "{\sf X}" features in their side-on views at energies \textit{beyond} the ILR, may also have a double boxy character. If populated, the extent of the inner boxiness along the major axis is about the same with that of the peanut supporting orbits in the side-on views. At any rate these orbits do not obscure the observation of the boxy orbits of the ILR region in the face-on views, as they contribute more to the surface density at the sides of the bar than to their central parts.

astro-ph.GA↗

arXiv2014

The phase-space of boxy-peanut and X-shaped bulges in galaxies I. Properties of non-periodic orbits

The investigation of the phase-space properties of structures encountered in a dynamical system is essential for understanding their formation and enhancement. In the present paper we explore the phase space in energy intervals where we have orbits that act as building blocks for boxy-peanut (b/p) and "{\sf X}-shaped" structures in rotating potentials of galactic type. We underline the significance of the rotational tori around the 3D families x1v1 and x1v1$^{\prime}$ that have been bifurcated from the planar x1 family. These tori play a multiple role: (i) They belong to quasi-periodic orbits that reinforce the local density. (ii) They act as obstacles for the diffusion of chaotic orbits and (iii) they attract a large number of chaotic orbits that become sticky to them. There are also bifurcations of unstable families (x1v2, x1v2$^{\prime}$). Their unstable asymptotic curves wind around the x1v1 and x1v1$^{\prime}$ tori generating orbits with hybrid morphologies between that of x1v1 and x1v2. In addition, a new family of multiplicity 2, called x1mul2, is found to be important for the peanut construction. Our work shows also that there are peanut-supporting orbits before the vertical ILR. Non-periodic orbits associated with the x1 family secure this contribution as well as the support of b/p structures at several other energy intervals. Non-linear phenomena associated with complex instability of single and double multiplicity families of periodic orbits show that these structures are not interrupted in regions where such orbits prevail. Depending on the main mechanism behind their formation, boxy bulges exhibit different morphological features. Finally our analysis indicates that "X" features shaped by orbits in the neighbourhood of x1v1 and x1v1$^{\prime}$ periodic orbits are pronounced only in side-on or nearly end-on views of the bar.

nlin.CD↗

arXiv2014

Analytical study of the structure of chaos near unstable points

In a 2D conservative Hamiltonian system there is a formal integral $Φ$ besides the energy H. This is not convergent near a stable periodic orbit, but it is convergent near an unstable periodic orbit. We explain this difference and we find the convergence radius along the asymptotic curves. In simple mappings this radius is infinite. This allows the theoretical calculation of the asymptotic curves and their intersections at homoclinic points. However in more complex mappings and in Hamiltonian systems the radius of convergence is in general finite and does not allow the theoretical calculation of any homoclinic point. Then we develop a method similar to analytic continuation, applicable in systems expressed in action-angle variables, that allows the calculation of the asymptotic curves to an arbitrary length. In this way we can study analytically the chaotic regions near the unstable periodic orbit and near its homoclinic points.

nlin.CD↗

arXiv2014

Analytical invariant manifolds near unstable points and the structure of chaos

It is known that the asymptotic invariant manifolds around an unstable periodic orbit in conservative systems can be represented by convergent series (Cherry 1926, Moser 1956, 1958, Giorgilli 2001). The unstable and stable manifolds intersect at an infinity of homoclinic points, generating a complicated homoclinic tangle. In the case of simple mappings it was found (Da Silva Ritter et al. 1987) that the domain of convergence of the formal series extends to infinity along the invariant manifolds. This allows in practice to study the homoclinic tangle using only series. However in the case of Hamiltonian systems, or mappings with a finite analyticity domain,the convergence of the series along the asymptotic manifolds is also finite. Here, we provide numerical indications that the convergence does not reach any homoclinic points. We discuss in detail the convergence problem in various cases and we find the degree of approximation of the analytical invariant manifolds to the real (numerical) manifolds as i) the order of truncation of the series increases, and ii) we use higher numerical precision in computing the coefficients of the series. Then we introduce a new method of series composition, by using action-angle variables, that allows the calculation of the asymptotic manifolds up to an a arbitrarily large extent. This is the first case of an analytic development that allows the computation of the invariant manifolds and their intersections in a Hamiltonian system for an extent long enough to allow the study of homoclinic chaos by analytical means.

nlin.CD↗

arXiv2014

The structure of phase space close to fixed points in a 4D symplectic map

We study the dynamics in the neighborhood of fixed points in a 4D symplectic map by means of the color and rotation method. We compare the results with the corresponding cases encountered in galactic type potentials and we find that they are in good agreement. The fact that the 4D phase space close to fixed points is similar to the 4D representations of the surfaces of section close to periodic orbits, indicates an archetypical 4D pattern for each kind of (in)stability, not only in 3D autonomous Hamiltonian systems with galactic type potentials but for a larger class of dynamical systems. This pattern is successfully visualized with the method we use in the paper.

nlin.CD↗