Unbounded Motion Research Articles

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

The Hénon–Heiles potential was first proposed as a simplified version of the gravitational potential experimented by a star in the presence of a galactic center. Currently, this system is considered a paradigm in dynamical systems because despite its simplicity exhibits a very complex dynamical behavior. In the present paper, we perform a series expansion up to the fifth-order of a potential with axial and reflection symmetries, which after some transformations, leads to a generalized Hénon–Heiles potential. Such new system is analyzed qualitatively in both regimes of bounded and unbounded motion via the Poincaré sections method and plotting the exit basins. On the other hand, the quantitative analysis is performed through the Lyapunov exponents and the basin entropy, respectively. We find that in both regimes the chaoticity of the system decreases as long as the test particle energy gets far from the critical energy. Additionally, we may conclude that despite the inclusion of higher order terms in the series expansion, the new system shows wider zones of regularity (islands) than the ones present in the Hénon–Heiles system.

Correlating the escape dynamics and the role of the normally hyperbolic invariant manifolds in a binary system of dwarf spheroidal galaxies

We elucidate the escape properties of stars moving in the combined gravitational field of a binary system of two interacting dwarf spheroidal galaxies. A galaxy model of three degrees of freedom is adopted for describing the dynamical properties of the Hamiltonian system. All the numerical values of the involved parameters are chosen having in mind the real binary system of the dwarf spheroidal galaxies NGC 147 and NGC 185. We distinguish between bounded (regular, sticky or chaotic) and escaping motion by classifying initial conditions of orbits in several types of two dimensional planes, considering only unbounded motion for several energy levels. We analyse the orbital structure of all types of two dimensional planes of initial conditions by locating the basins of escape and also by measuring the corresponding escape time of the orbits. Furthermore, the properties of the normally hyperbolic invariant manifolds (NHIMs), located in the vicinity of the index-1 saddle points L1, L2, and L3, are also investigated. These manifolds are of great importance, as they control the flow of stars (between the two galaxies and towards the exterior region) over the different saddle points. In addition, bifurcation diagrams of the Lyapunov periodic orbits as well as restrictions of the Poincaré map to the NHIMs are presented for revealing the dynamics in the neighbourhood of the saddle points. Comparison between the current outcomes and previous related results is also made.

Use of Continuous Traits Can Improve Morphological Phylogenetics

The recent surge in enthusiasm for simultaneously inferring relationships from extinct and extant species has reinvigorated interest in statistical approaches for modeling morphological evolution. Current statistical methods use the Mk model to describe substitutions between discrete character states. Although representing a significant step forward, the Mk model presents challenges in biological interpretation, and its adequacy in modeling morphological evolution has not been well explored. Another major hurdle in morphological phylogenetics concerns the process of character coding of discrete characters. The often subjective nature of discrete character coding can generate discordant results that are rooted in individual researchers' subjective interpretations. Employing continuous measurements to infer phylogenies may alleviate some of these issues. Although not widely used in the inference of topology, models describing the evolution of continuous characters have been well examined, and their statistical behavior is well understood. Also, continuous measurements avoid the substantial ambiguity often associated with the assignment of discrete characters to states. I present a set of simulations to determine whether use of continuous characters is a feasible alternative or supplement to discrete characters for inferring phylogeny. I compare relative reconstruction accuracy by inferring phylogenies from simulated continuous and discrete characters. These tests demonstrate significant promise for continuous traits by demonstrating their higher overall accuracy as compared to reconstruction from discrete characters under Mk when simulated under unbounded Brownian motion, and equal performance when simulated under an Ornstein-Uhlenbeck model. Continuous characters also perform reasonably well in the presence of covariance between sites. I argue that inferring phylogenies directly from continuous traits may be benefit efforts to maximize phylogenetic information in morphological data sets by preserving larger variation in state space compared to many discretization schemes. I also suggest that the use of continuous trait models in phylogenetic reconstruction may alleviate potential concerns of discrete character model adequacy, while identifying areas that require further study in this area. This study provides an initial controlled demonstration of the efficacy of continuous characters in phylogenetic inference.

Control of the motion of a triaxial ellipsoid in a fluid using rotors

The motion of a body shaped as a triaxial ellipsoid and controlled by the rotation of three internal rotors is studied. It is proved that the motion is controllable with the exception of a few particular cases. Partial solutions whose combinations enable an unbounded motion in any arbitrary direction are constructed.

$$(2+1)$$ ( 2 + 1 ) -Dimensional charged black holes with scalar hair in Einstein–Power–Maxwell Theory

We obtain an exact static solution to Einstein-Power-Maxwell (EPM) theory in $(2+1)$ dimensional AdS spacetime, in which the scalar field couples to gravity in a non-minimal way. After considering the scalar potential, a stable system leads to a constraint on the power parameter $k$ of Maxwell field. The solution contains a curvature singularity at the origin and is non-conformally flat. The horizon structures are identified, which indicates the physically acceptable lower bound of mass in according to the existence of black hole solutions. Especially for the cases with $k>1$, the lower bound is negative, thus there exist scalar black holes with negative mass. The null geodesics in this spacetime are also discussed in detail. They are divided into five models, which are made up of the cases with the following geodesic motions: no-allowed motion, the circular motion, the elliptic motion and the unbounded spiral motion.

An overview of the escape dynamics in the Hénon–Heiles Hamiltonian system

The aim of this work is to revise but also explore even further the escape dynamics in the H\'{e}non-Heiles system. In particular, we conduct a thorough and systematic numerical investigation distinguishing between trapped (ordered and chaotic) and escaping orbits, considering only unbounded motion for several energy levels. It is of particular interest, to locate the basins of escape towards the different escape channels and relate them with the corresponding escape periods of the orbits. In order to elucidate the escape process we conduct a thorough investigation in several types of two-dimensional planes and also in a three-dimensional subspace of the entire four-dimensional phase space. We classify extensive samples of orbits by integrating numerically the equations of motion as well as the variational equations. In an attempt to determine the regular or chaotic nature of trapped motion, we apply the SALI method, as an accurate chaos detector. It was found, that in all studied cases regions of non-escaping orbits coexist with several basins of escape. Most of the current outcomes have been compared with previous related work.

Control of the motion of a helical body in a fluid using rotors

This paper is concerned with the motion of a helical body in an ideal fluid, which is controlled by rotating three internal rotors. It is proved that the motion of the body is always controllable by means of three rotors with noncoplanar axes of rotation. A condition whose satisfaction prevents controllability by means of two rotors is found. Control actions that allow the implementation of unbounded motion in an arbitrary direction are constructed. Conditions under which the motion of the body along an arbitrary smooth curve can be implemented by rotating the rotors are presented. For the optimal control problem, equations of sub-Riemannian geodesics on SE(3) are obtained.

Unravelling the escape dynamics and the nature of the normally hyperbolic invariant manifolds in tidally limited star clusters

The escape mechanism of orbits in a star cluster rotating around its parent galaxy in a circular orbit is investigated. A three degrees of freedom model is used for describing the dynamical properties of the Hamiltonian system. The gravitational field of the star cluster is represented by a smooth and spherically symmetric Plummer potential. We distinguish between ordered and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels. The Smaller Alignment Index (SALI) method is used for determining the regular or chaotic nature of the orbits. The basins of escape are located and they are also correlated with the corresponding escape time of the orbits. Areas of bounded regular or chaotic motion and basins of escape were found to coexist in the $(x,z)$ plane. The properties of the normally hyperbolic invariant manifolds (NHIMs), located in the vicinity of the index-1 Lagrange points $L_1$ and $L_2$, are also explored. These manifolds are of paramount importance as they control the flow of stars over the saddle points, while they also trigger the formation of tidal tails observed in star clusters. Bifurcation diagrams of the Lyapunov periodic orbits as well as restrictions of the Poincar\'e map to the NHIMs are deployed for elucidating the dynamics in the neighbourhood of the saddle points. The extended tidal tails, or tidal arms, formed by stars with low velocity which escape through the Lagrange points are monitored. The numerical results of this work are also compared with previous related work.

Safe basins for a nonlinear oscillator with ramped forcing

A nonlinear oscillator is studied in the presence of external forcing for which the amplitude initially depends on time. The focus is on the sizes of the basins of attraction which do not lead to unbounded motions, collectively termed the ‘safe basin’. Direct comparisons are drawn between the regime of constant forcing amplitude and that where the forcing amplitude initially depends on time. In the process, questions from previous literature are answered and previously unexplained phenomena are understood. Furthermore, we witness a new phenomenon, not previously observed for the system studied.

SIMULATION OF HYBRID ELECTRIC VEHICLE BASED ON A SERIES DRIVE TRAIN LAYOUT

This paper provided a validated modeling and a simulation of a 6 degree freedom vehicle longitudinal model and drive-train component in a series hybrid electric vehicle. The 6-DOF vehicle dynamics model consisted of tire subsystems, permanent magnet synchronous motor which acted as the prime mover coupled with an automatic transmission, hydraulic brake subsystem, battery subsystem, alternator subsystem and internal combustion engine to supply the rotational input to the alternator. A speed and torque tracking control systems of the electric power train were developed to make sure that the power train was able to produce the desired throttle torque in accelerating the vehicle. A human-in-the-loop-simulation was utilized as a mechanism to evaluate the effectiveness of the proposed hybrid electric vehicle. The proposed simulation was used as the preliminary result in identifying the capability of the vehicle in terms of the maximum speed produced by the vehicle and the capability of the alternator to recharge the battery. Several tests had been done during the simulation, namely sudden acceleration, acceleration and braking test and unbounded motion. The results of the simulation showed that the proposed hybrid electric vehicle can produce a speed of up to 70 km/h with a reasonable charging rate to the battery. The findings from this study can be considered in terms of design, optimization and implementation in a real vehicle.

Interaction of a two-layer vortex pair with a submerged cylindrical obstacle in a two layer rotating fluid

We consider the dynamics of a two-layer compensated vortex pair (heton) interacting with a submerged cylindrical obstacle of small height located in the lower layer of a two-layer fluid in the f-plane. The pair consists of two counter-rotating vortices of equal strengths each located in different layers of the two-layer rotating fluid. We make use of two approaches. The first one considers a model of point vortices, and the second one assumes the vortices as finite-core vorticity patches analyzed by means of contour dynamics techniques. The point vortex model features two regimes of the pair’s motion: an unbounded motion as the pair advances to infinity after being deflected by the cylindrical obstacle and an oscillatory motion inside a bounded region near the cylindrical obstacle. The oscillations, in turn, are of two types. The first corresponds to a finite yet unpredictable number of vortex revolutions near the cylinder, and the second results in an infinite number of revolutions. By exploiting contour dynamics techniques, we obtain very similar unbounded dynamics of a distributed vorticity heton given relatively strong stratification. An important feature of this dynamics is that the contours associated with the pair hold almost unperturbed signifying sufficient stability. By decreasing stratification, we observe complicated dynamics involving vorticity redistribution and vortex splitting.

Fugitive stars in active galaxies

We investigate in detail the escape dynamics in an analytical gravitational model which describes the motion of stars in a quasar galaxy with a disk and a massive nucleus. We conduct a thorough numerical analysis distinguishing between regular and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels. In order to distinguish safely and with certainty between ordered and chaotic motion we apply the Smaller ALingment Index (SALI) method. It is of particular interest to locate the escape basins through the openings around the collinear Lagrangian points $L_1$ and $L_2$ and relate them with the corresponding spatial distribution of the escape times of the orbits. Our exploration takes place both in the configuration $(x,y)$ and in the phase $(x,\dot{x})$ space in order to elucidate the escape process as well as the overall orbital properties of the galactic system. Our numerical analysis reveals the strong dependence of the properties of the considered escape basins with the total orbital energy, with a remarkable presence of fractal basin boundaries along all the escape regimes. We hope our outcomes to be useful for a further understanding of the escape mechanism in active galaxy models.

Nonlinear thermal dynamic buckling of FGM beams

Abstract Based on the nonlinear dynamic analysis, dynamic buckling and imperfection sensitivity of the FGM Timoshenko beam subjected to sudden uniform temperature rise are studied. Initial geometric imperfection of the beam is also taken into account. It is assumed that during deformation the beam is resting over a conventional three-parameter elastic foundation with softening/hardening cubic non-linearity. The analysis is performed with considering temperature dependency assumption of each thermo-mechanical property of the FGM beam. The governing nonlinear dynamic equations are derived based on the generalized Hamilton principle. In the spatial approximation of the problem, a set of ordinary differential equations in time is obtained by the conventional multi-term Ritz method. These equations are converted into a set of algebraic equations by utilizing the Newmark family of time approximation scheme. The obtained non-linear algebraic equations are solved via the well known Newton–Raphson iterative scheme. The Budiansky–Roth criterion is used to detect the unbounded motion type of dynamic buckling. Results reveal that for beams with stable post-buckling equilibrium path, no dynamic buckling occurs according to the Budiansky–Roth criterion. However, dynamic buckling may occur for the FGM beams resting on sufficiently stiff softening elastic foundation due to their unstable post-buckling equilibrium paths.

Comparing the escape dynamics in tidally limited star cluster models

The aim of this work is to compare the orbital dynamics in three different models describing the properties of a star cluster rotating around its parent galaxy in a circular orbit. In particular, we use the isochrone and the Hernquist potentials to model the spherically symmetric star cluster, and we compare our results with the corresponding ones of a previous work in which the Plummer model was applied for the same purpose. Our analysis takes place both in the configuration (x, y) and in the phase |$(x,\dot{x})$| space in order to elucidate the escape process as well as the overall orbital properties of the tidally limited star cluster. We restrict our investigation into two dimensions, and we conduct a thorough numerical analysis distinguishing between ordered and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels above the critical escape energy. It is of particular interest to determine the escape basins towards the two exit channels (near the Lagrangian points L1 and L2) and relate them with the corresponding escape times of the orbits.

Escape dynamics in a binary system of interacting galaxies

The escape dynamics in an analytical gravitational model which describes the motion of stars in a binary system of interacting dwarf spheroidal galaxies is investigated in detail. We conduct a numerical analysis distinguishing between regular and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels. In order to distinguish safely and with certainty between ordered and chaotic motion, we apply the Smaller ALingment Index (SALI) method. It is of particular interest to locate the escape basins through the openings around the collinear Lagrangian points L1 and L2 and relate them with the corresponding spatial distribution of the escape times of the orbits. Our exploration takes place both in the configuration (x, y) and in the phase (x,x˙) space in order to elucidate the escape process as well as the overall orbital properties of the galactic system. Our numerical analysis reveals the strong dependence of the properties of the considered escape basins with the total orbital energy, with a remarkable presence of fractal basin boundaries along all the escape regimes. It was also observed that for energy levels close to the critical escape energy the escape rates of orbits are large, while for much higher values of energy most of the orbits have low escape periods or they escape immediately to infinity. We hope our outcomes to be useful for a further understanding of the escape mechanism in binary galaxy models.

Periodic and unbounded motions in asymmetric oscillators at resonance

We consider the existence of periodic and unbounded solutions for the asymmetric oscillatorx″+ax+−bx−+g(x)=p(t), where x+=max⁡{x,0}, x−=max⁡{−x,0}, a and b are two positive constants, p(t) is a 2π-periodic smooth function and g(x) satisfies lim|x|→+∞⁡x−1g(x)=0. We have proved previously that the boundedness of all the solutions and the existence of unbounded solutions have a close relation to the interaction of some well-defined functions Φp(θ) and Λ(h). In this paper, we consider some critical cases and obtain some new sufficient conditions for the existence of 2π-periodic and unbounded solutions. In particular, the obtained periodic solutions are new and cannot be deduced from Landesman–Lazer's and Fabry–Fonda's existence conditions. In contrast with many existing results, the function g(x) may be unbounded or oscillatory without asymptotic limits.

A solution procedure based on the Ateb function for a two-degree-of-freedom oscillator

In this paper vibration of a two mass system with two degrees of freedom is considered. Two equal harmonic oscillators are coupled with a strong nonlinear viscoelastic connection. Mathematical model of the system is two coupled second-order strong nonlinear differential equations. Introducing new variables the system transforms into two uncoupled equations: one of them is linear and the other with a strong nonlinearity. In the paper a method for solving the strong nonlinear equation is developed. Based on the exact solution of a pure nonlinear differential equation, we assumed a perturbed version of the solution with time variable parameters. Due to the fact that the solution is periodical, the averaging procedure is introduced. As a special case vibrations of harmonic oscillators with fraction order nonlinear connection are considered. Depending on the order and coefficient of nonlinearities bounded and unbounded motion of masses is determined. Besides, the conditions for steady-state periodical solution are discussed. The procedure given in the paper is applied for investigation of the vibration of a vocal cord, which is modeled with two harmonic oscillators with strong nonlinear fraction order viscoelastic connection. Using the experimental data for the vocal cord the parameters for the steady-state solution which describes the flexural vibration of the vocal cord is analyzed. The influence of the order of nonlinearity on the amplitude and frequency of vibration of the vocal cord is obtained. The analytical results are close to those obtained experimentally.

Periodic, quasi-periodic and unbounded solutions of radially symmetric systems with repulsive singularities at resonance

In this paper, we are concerned with periodic solutions, quasi-periodic solutions and unbounded solutions for radially symmetric systems with singularities at resonance, which are 2π-periodic in time. The method is based on the qualitative analysis of Poincare map with action-angle variables. The existence of infinitely many periodic and quasi-periodic solutions or unbounded motions depends on the oscillatory properties of a certain function.

Escape dynamics and fractal basin boundaries in Seyfert galaxies

The escape dynamics in a simple analytical gravitational model which describes the motion of stars in a Seyfert galaxy is investigated in detail. We conduct a thorough numerical analysis distinguishing between regular and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels. In order to distinguish safely and with certainty between ordered and chaotic motion, we apply the Smaller ALingment Index (SALI) method. It is of particular interest to locate the escape basins through the openings around the collinear Lagrangian points $L_1$ and $L_2$ and relate them with the corresponding spatial distribution of the escape times of the orbits. Our exploration takes place both in the physical $(x,y)$ and in the phase $(x,\dot{x})$ space in order to elucidate the escape process as well as the overall orbital properties of the galactic system. Our numerical analysis reveals the strong dependence of the properties of the considered escape basins with the total orbital energy, with a remarkable presence of fractal basin boundaries along all the escape regimes. It was also observed, that for energy levels close to the critical escape energy the escape rates of orbits are large, while for much higher values of energy most of the orbits have low escape periods or they escape immediately to infinity. We also present evidence obtained through numerical simulations that our model can describe the formation and the evolution of the observed spiral structure in Seyfert galaxies. We hope our outcomes to be useful for a further understanding of the escape mechanism in galaxies with active nuclei.

Revealing the escape mechanism of three-dimensional orbits in a tidally limited star cluster

The aim of this work is to explore the escape process of three-dimensional orbits in a star cluster rotating around its parent galaxy in a circular orbit. The gravitational field of the cluster is represented by a smooth, spherically symmetric Plummer potential, while the tidal approximation was used to model the steady tidal field of the galaxy. We conduct a thorough numerical analysis distinguishing between regular and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels. It is of particular interest to locate the escape basins towards the two exit channels and relate them with the corresponding escape times of the orbits. For this purpose, we split our investigation into three cases depending on the initial value of the $z$ coordinate which was used for launching the stars. The most noticeable finding is that the majority of stars initiated very close to the primary $(x,y)$ plane move in chaotic orbits and they remain trapped for vast time intervals, while orbits with relatively high values of $z_0$ on the other hand, form well-defined basins of escape. It was also observed, that for energy levels close to the critical escape energy the escape rates of orbits are large, while for much higher values of energy most of the orbits have low escape periods or they escape immediately to infinity. We hope our outcomes to be useful for a further understanding of the dissolution process and the escape mechanism in open star clusters.