scholarly journals Simple Periodic Orbits in Elliptical Galaxies Modelled by Hamiltonians in 1-1-1 Resonance

1999 ◽  
Vol 172 ◽  
pp. 411-412
Author(s):  
J. Palacián ◽  
P. Yanguas ◽  
S. Ferrer
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Harmonic Oscillator ◽  
Periodic Orbits ◽  
Homogeneous Polynomial ◽  
Three Dimensional ◽  
Discrete Symmetry ◽  
Elliptical Galaxies ◽  
Isotropic Harmonic Oscillator

AbstractWe consider elliptical galactic models, whose dynamical system consists of a three-dimensional isotropic harmonic oscillator plus a potential given by a homogeneous polynomial of degree four with an additional discrete symmetry. We identify families of simple periodic orbits by studying the reduced phase space.

scholarly journals Complexity of a Microblogging Social Network in the Framework of Modern Nonlinear Science

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Andrey Dmitriev ◽  
Vasily Kornilov ◽  
Svetlana Maltseva
Keyword(s):  
Dynamical System ◽  
Social Network ◽  
Phase Space ◽  
Adiabatic Approximation ◽  
Three Dimensional ◽  
Evolutionary Model ◽  
Random Dynamical System ◽  
New Paradigm ◽  
Dimensional Phase Space ◽  
Nonlinear Science

Recent developments in nonlinear science have caused the formation of a new paradigm called the paradigm of complexity. The self-organized criticality theory constitutes the foundation of this paradigm. To estimate the complexity of a microblogging social network, we used one of the conceptual schemes of the paradigm, namely, the system of key signs of complexity of the external manifestations of the system irrespective of its internal structure. Our research revealed all the key signs of complexity of the time series of a number of microposts. We offer a new model of a microblogging social network as a nonlinear random dynamical system with additive noise in three-dimensional phase space. Implementations of this model in the adiabatic approximation possess all the key signs of complexity, making the model a reasonable evolutionary model for a microblogging social network. The use of adiabatic approximation allows us to model a microblogging social network as a nonlinear random dynamical system with multiplicative noise with the power-law in one-dimensional phase space.

scholarly journals Dynamical system analysis of quintessence models with exponential potential — Revisited

2019 ◽  
Vol 34 (09) ◽  
pp. 1950069
Author(s):  
A. Savaş Arapoğlu ◽  
A. Emrah Yükselci
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Equation Of State ◽  
System Analysis ◽  
Three Dimensional ◽  
State Parameter ◽  
Late Time ◽  
Exponential Potential ◽  
Effective Equation ◽  
Equation Of State Parameter

Dynamical system analysis of a universe model which contains matter, radiation and quintessence with exponential potential, [Formula: see text], is studied in the light of recent observations and the tensions between different datasets. The three-dimensional phase space is constructed by the energy density parameters and all the critical points of the model with their physical meanings are investigated. This approach provides an easy way of comparing the model directly with the observations. We consider a solution that is compatible with observations and is continuous in the phase space in both directions of time, past and future. Although in many studies of late-time acceleration, the radiation is neglected, here we consider all components together and this makes the calculated effective equation of state parameter more realistic. Additionally, a relation between potential parameter, [Formula: see text], and the value of quintessence equation of state parameter, [Formula: see text], today is found by using numerical analysis. We conclude that [Formula: see text] has to be small in order to explain the current accelerated phase of the universe and this result can be seen directly from the relation we obtain. Finally, we compare the usual dynamical system approach with the approach that we follow in this paper.

scholarly journals Building CX peanut-shaped disk galaxy profiles

2018 ◽  
Vol 612 ◽  
pp. A114 ◽  
Author(s):  
P. A. Patsis ◽  
M. Harsoula
Keyword(s):  
Phase Space ◽  
Periodic Orbits ◽  
Three Dimensional ◽  
Resonance Region ◽  
Space Structure ◽  
Jacobi Constant ◽  
Dynamical Mechanism ◽  
Standard Case ◽  
New Model ◽  
Stable Periodic

Context. We present and discuss the orbital content of a rather unusual rotating barred galaxy model, in which the three-dimensional (3D) family, bifurcating from x1 at the 2:1 vertical resonance with the known “frown-smile” side-on morphology, is unstable. Aims. Our goal is to study the differences that occur in the phase space structure at the vertical 2:1 resonance region in this case, with respect to the known, well studied, standard case, in which the families with the frown-smile profiles are stable and support an X-shaped morphology. Methods. The potential used in the study originates in a frozen snapshot of an N-body simulation in which a fast bar has evolved. We follow the evolution of the vertical stability of the central family of periodic orbits as a function of the energy (Jacobi constant) and we investigate the phase space content by means of spaces of section. Results. The two bifurcating families at the vertical 2:1 resonance region of the new model change their stability with respect to that of most studied analytic potentials. The structure in the side-on view that is directly supported by the trapping of quasi-periodic orbits around 3D stable periodic orbits has now an infinity symbol (i.e. ∞-type) profile. However, the available sticky orbits can reinforce other types of side-on morphologies as well. Conclusions. In the new model, the dynamical mechanism of trapping quasi-periodic orbits around the 3D stable periodic orbits that build the peanut, supports the ∞-type profile. The same mechanism in the standard case supports the X shape with the frown-smile orbits. Nevertheless, in both cases (i.e. in the new and in the standard model) a combination of 3D quasi-periodic orbits around the stable x1 family with sticky orbits can support a profile reminiscent of the shape of the orbits of the 3D unstable family existing in each model.

scholarly journals 1/1 resonant periodic orbits in three dimensional planetary systems

2014 ◽  
Vol 9 (S310) ◽  
pp. 82-83 ◽  
Author(s):  
Kyriaki I. Antoniadou ◽  
George Voyatzis ◽  
Harry Varvoglis
Keyword(s):  
Phase Space ◽  
Linear Stability ◽  
Periodic Orbits ◽  
Orbital Motion ◽  
Three Dimensional ◽  
Planetary Systems ◽  
Planetary Mass ◽  
Construct Maps ◽  
Mutual Inclination ◽  
Planar Family

AbstractWe study the dynamics of a two-planet system, which evolves being in a 1/1 mean motion resonance (co-orbital motion) with non-zero mutual inclination. In particular, we examine the existence of bifurcations of periodic orbits from the planar to the spatial case. We find that such bifurcations exist only for planetary mass ratios $\rho=\frac{m_2}{m_1}<0.0205$. For ρ in the interval 0<ρ<0.0205, we compute the generated families of spatial periodic orbits and their linear stability. These spatial families form bridges, which start and end at the same planar family. Along them the mutual planetary inclination varies. We construct maps of dynamical stability and show the existence of regions of regular orbits in phase space.

KNOTTED PERIODIC SOLUTIONS OF A LINEAR NONAUTONOMOUS SYSTEM AND SOME RELATED THREE-DIMENSIONAL FLOWS

2012 ◽  
Vol 22 (11) ◽  
pp. 1250280
Author(s):  
JIBIN LI ◽  
XIAOHUA ZHAO
Keyword(s):  
Phase Space ◽  
Periodic Solutions ◽  
Periodic Orbits ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
Distinct Type ◽  
Frequency Parameter ◽  
Bounded Region ◽  
Nonautonomous Systems ◽  
Dimensional Phase Space

This paper considers a three-dimensional linear nonautonomous systems. It shows that for every integer frequency parameter value, this system has a distinct type of knotted periodic solutions, which lie in a bounded region of R3. Exact explicit parametric representations of the knotted periodic solutions are given. By using these parametric representations, two series of three-dimensional flows are constructed, such that these three-dimensional autonomous systems have knotted periodic orbits in the three-dimensional phase space.

scholarly journals Integrable zero-Hopf singularities and three-dimensional centres

2017 ◽  
Vol 148 (2) ◽  
pp. 327-340
Author(s):  
Isaac A. García
Keyword(s):  
Phase Space ◽  
Normal Form ◽  
Periodic Orbits ◽  
Parameter Space ◽  
Three Dimensional ◽  
Quadratic Polynomial ◽  
Affine Variety ◽  
Higher Dimensional ◽  
Dimensional Version ◽  
Poincaré Return Map

In this paper we show that the well-known Poincaré–Lyapunov non-degenerate analytic centre problem in the plane and its higher-dimensional version, expressed as the three-dimensional centre problem at the zero-Hopf singularity, have a lot of common properties. In both cases the existence of a neighbourhood of the singularity in the phase space completely foliated by periodic orbits (including equilibria) is characterized by the fact that the system is analytically completely integrable. Hence its Poincaré–Dulac normal form is analytically orbitally linearizable. There also exists an analytic Poincaré return map and, when the system is polynomial and parametrized by its coefficients, the set of systems with centres corresponds to an affine variety in the parameter space of coefficients. Some quadratic polynomial families are considered.

THREE-DIMENSIONAL ISOTROPIC PSEUDO-GAUSSIAN OSCILLATORS

2009 ◽  
Vol 20 (07) ◽  
pp. 1103-1111 ◽  
Author(s):  
ION I. COTĂESCU ◽  
PAUL GRĂVILĂ ◽  
MARIUS PAULESCU
Keyword(s):  
Asymptotic Behavior ◽  
Numerical Methods ◽  
Harmonic Oscillator ◽  
Energy Levels ◽  
Three Dimensional ◽  
Finite Energy ◽  
Energy Spectra ◽  
Isotropic Harmonic Oscillator ◽  
Quantum Models

A family of isotropic three-dimensional quantum models governed by isotropic pseudo-Gaussian potentials is proposed. These potentials are defined to have a Gaussian asymptotic behavior but approaching to the potential of the isotropic harmonic oscillator when x → 0. These models may have finite energy spectra with approximately equidistant energy levels that can be calculated using efficient numerical methods based on generating functionals.

scholarly journals Characteristics of Periodic Orbits in Elliptical Galaxies

1983 ◽  
Vol 74 ◽  
pp. 271-274
Author(s):  
N. Caranicolas
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Degrees Of Freedom ◽  
Basic Characteristic ◽  
Elliptical Galaxies ◽  
Resonance Case ◽  
Two Degrees Of Freedom ◽  
Exact Resonance ◽  
Near Resonance ◽  
Families Of Periodic Orbits

AbstractThe properties of the characteristic curves of several families of periodic orbits, in a conservative dynamical system of two degrees of freedom, symmetric with respect to both axes, are reviewed. The two main types of families are presented. One sees that the pattern of the characteristics in the exact resonance case is similar to that of the near resonance case except for the basic characteristic . The form of the characteristics can be found theoretically by means of the second integral.

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