scholarly journals Trapped and Escaping Orbits in an Axially Symmetric Galactic-Type Potential

2012 ◽  
Vol 29 (2) ◽  
pp. 161-173 ◽  
Author(s):  
Euaggelos E. Zotos
Keyword(s):  
Dynamical System ◽  
Characteristic Exponent ◽  
Axially Symmetric ◽  
Surface Of Section ◽  
Lyapunov Characteristic Exponent ◽  
Average Value ◽  
Chaotic Orbits ◽  
Galactic Model ◽  
Escaping Orbits ◽  
Type Potential

AbstractIn the present article, we investigate the behavior of orbits in a time-independent axially symmetric galactic-type potential. This dynamical model can be considered to describe the motion in the central parts of a galaxy, for values of energies larger than the energy of escape. We use the classical surface-of-section method in order to visualize and interpret the structure of the phase space of the dynamical system. Moreover, the Lyapunov characteristic exponent is used in order to make an estimation of the degree of chaoticity of the orbits in our galactic model. Our numerical calculations suggest that in this galactic-type potential there are two kinds of orbits: (i) escaping orbits and (ii) trapped orbits, which do not escape at all. Furthermore, a large number of orbits of the dynamical system display chaotic motion. Among the chaotic orbits, there are orbits that escape quickly and also orbits that remain trapped for vast time intervals. When the value of a test particle's energy slightly exceeds the energy of escape, the number of trapped regular orbits increases as the value of the angular momentum increases. Therefore, the extent of the chaotic regions observed in the phase plane decreases as the energy value increases. Moreover, we calculate the average value of the escape period of chaotic orbits and try to correlate it with the value of the energy and also with the maximum value of the z component of the orbits. In addition, we find that the value of the Lyapunov characteristic exponent corresponding to each chaotic region for different values of energy increases exponentially as the energy increases. Some theoretical arguments are presented in order to support the numerically obtained outcomes.

Non-Uniform Chaotic Dynamics with Implications to Information Processing

1983 ◽  
Vol 38 (11) ◽  
pp. 1157-1169 ◽  
Author(s):  
J. S. Nicolis ◽  
G. Meyer-Kress ◽  
G. Haubs
Keyword(s):  
Information Processing ◽  
Characteristic Exponent ◽  
Quantitative Measure ◽  
External Noise ◽  
Strange Attractors ◽  
Pattern Perception ◽  
Small Subset ◽  
Chaotic Flows ◽  
Lyapunov Characteristic Exponent ◽  
High Degree

We study a new parameter - the "Non-Uniformity Factor" (NUF) -, which we have introduced in [1]. by way of estimating and comparing the deviation from average behavior (expressed by such factors as the Lyapunov characteristic exponent(s) and the information dimension) in various strange attractors (discrete and chaotic flows). Our results show for certain values of the control parameters the inadequacy of the above averaging properties in representing what is actually going on - especially when the strange attractors are employed as dynamical models for information processing and pattern recognition. In such applications (like for example visual pattern perception or communication via a burst-error channel) the high degree of adherence of the processor to a rather small subset of crucial features of the pattern under investigation or the flow, has been documented experimentally: Hence the weakness of concepts such as the entropy in giving in such cases a quantitative measure of the information transaction between the pattern and the processor. We finally investigate the influence of external noise in modifying the NUF

scholarly journals The Orbital Dynamics of Synchronous Satellites: Irregular Motions in the 2 : 1 Resonance

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Jarbas Cordeiro Sampaio ◽  
Rodolpho Vilhena de Moraes ◽  
Sandro da Silva Fernandes
Keyword(s):  
Dynamical System ◽  
Numerical Integration ◽  
Lyapunov Exponents ◽  
Orbital Dynamics ◽  
Zonal Harmonics ◽  
Chaotic Orbits ◽  
The Earth ◽  
Tesseral Harmonics ◽  
Final System

The orbital dynamics of synchronous satellites is studied. The 2 : 1 resonance is considered; in other words, the satellite completes two revolutions while the Earth completes one. In the development of the geopotential, the zonal harmonicsJ20andJ40and the tesseral harmonicsJ22andJ42are considered. The order of the dynamical system is reduced through successive Mathieu transformations, and the final system is solved by numerical integration. The Lyapunov exponents are used as tool to analyze the chaotic orbits.

scholarly journals ORBITS IN A NON-KERR DYNAMICAL SYSTEM

2011 ◽  
Vol 21 (08) ◽  
pp. 2261-2277 ◽  
Author(s):  
G. CONTOPOULOS ◽  
G. LUKES-GERAKOPOULOS ◽  
T. A. APOSTOLATOS
Keyword(s):  
Angular Momentum ◽  
Outer Region ◽  
Kerr Black Hole ◽  
Compact Object ◽  
Kerr Metric ◽  
Integrable Case ◽  
Closed Curves ◽  
Surface Of Section ◽  
Fundamental Frequencies ◽  
Chaotic Orbits

We study the orbits in a Manko–Novikov type metric (MN) which is a perturbed Kerr metric. There are periodic, quasi-periodic, and chaotic orbits, which are found in configuration space and on a surface of section for various values of the energy E and the z-component of the angular momentum Lz. For relatively large Lz there are two permissible regions of nonplunging motion bounded by two closed curves of zero velocity (CZV), while in the Kerr metric there is only one closed CZV of nonplunging motion. The inner permissible region of the MN metric contains mainly chaotic orbits, but it contains also a large island of stability. When Lz decreases, the two permissible regions join and chaos increases. Below a certain value of Lz, most orbits escape inwards and plunge through the horizon. On the other hand, as the energy E decreases (for fixed Lz) the outer permissible region shrinks and disappears. In the inner permissible region, chaos increases and for sufficiently small E most orbits are plunging. We find the positions of the main periodic orbits as functions of Lz and E, and their bifurcations. Around the main periodic orbit of the outer region, there are islands of stability that do not appear in the Kerr metric (integrable case). In a realistic binary system, because of the gravitational radiation, the energy E and the angular momentum Lz of an inspiraling compact object decrease and therefore the orbit of the object is nongeodesic. In fact, in an extreme mass ratio inspiraling (EMRI) system the energy E and the angular momentum Lz decrease adiabatically and therefore the motion of the inspiraling object is characterized by the fundamental frequencies which are drifting slowly in time. In the Kerr metric, the ratio of the fundamental frequencies changes strictly monotonically in time. However, in the MN metric when an orbit is trapped inside an island the ratio of the fundamental frequencies remains constant for some time. Hence, if such a phenomenon is observed this will indicate that the system is nonintegrable and therefore the central object is not a Kerr black hole.

NONLINEAR DYNAMICS AND CHAOS PART I: A GEOMETRICAL APPROACH

1998 ◽  
Vol 2 (4) ◽  
pp. 505-532 ◽  
Author(s):  
Alfredo Medio
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamical Systems ◽  
Characteristic Exponent ◽  
Chaotic Attractors ◽  
Modern Theory ◽  
Geometrical Approach ◽  
Lyapunov Characteristic Exponent ◽  
Geometrical Properties ◽  
Nonlinear Dynamics And Chaos ◽  
Different Types

This paper is the first part of a two-part survey reviewing some basic concepts and methods of the modern theory of dynamical systems. The survey is introduced by a preliminary discussion of the relevance of nonlinear dynamics and chaos for economics. We then discuss the dynamic behavior of nonlinear systems of difference and differential equations such as those commonly employed in the analysis of economically motivated models. Part I of the survey focuses on the geometrical properties of orbits. In particular, we discuss the notion of attractor and the different types of attractors generated by discrete- and continuous-time dynamical systems, such as fixed and periodic points, limit cycles, quasiperiodic and chaotic attractors. The notions of (noninteger) fractal dimension and Lyapunov characteristic exponent also are explained, as well as the main routes to chaos.

scholarly journals Differentiability with respect to parameters of average values in probabilistic contracting dynamical systems

1990 ◽  
Vol 10 (3) ◽  
pp. 599-610 ◽  
Author(s):  
W. Douglas Withers
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Harmonic Function ◽  
Invariant Measure ◽  
Compact Subset ◽  
Julia Set ◽  
Spatial Variable ◽  
External Parameter ◽  
Average Value ◽  
Complex Functions

AbstractWe consider a dynamical system consisting of a compact subset of RN or CN with several contracting maps chosen with prescribed probabilities, which may depend on position. We show that if the maps and the probabilities are Cl+α functions of the spatial variable and an external parameter, then the average value of a Cl+α function is a differentiate function of the parameter. One implication of this theorem is that for certain families of complex functions dependent on a parameter the reciprocal of the dimension of an invariant measure on the Julia set is a harmonic function of the parameter.

scholarly journals Alternative Detection of n = 1 Modes Slowing Down on ASDEX Upgrade

2020 ◽  
Vol 10 (21) ◽  
pp. 7891
Author(s):  
Emmanuele Peluso ◽  
Riccardo Rossi ◽  
Andrea Murari ◽  
Pasqualino Gaudio ◽  
Michela Gelfusa ◽  
...  
Keyword(s):  
Weighted Average ◽  
Learning Approaches ◽  
Axially Symmetric ◽  
Average Value ◽  
Slowing Down ◽  
Mhd Instabilities ◽  
Physically Based ◽  
Magnetic Perturbations ◽  
Edge Localized Modes ◽  
Alternative Detection

Disruptions in tokamaks are very often associated with the slowing down of magneto-hydrodynamic (MHD) instabilities and their subsequent locking to the wall. To improve the understanding of the chain of events ending with a disruption, a statistically robust and physically based criterion has been devised to track the slowing down of modes with toroidal mode numbers n = 1 and mostly poloidal mode number m = 2, providing an alternative and earlier detection tool compared to simple threshold based indicators. A database of 370 discharges of axially symmetric divertor experiment—upgrade (AUG) has been studied and results compared with other indicators used in real time. The estimator is based on a weighted average value of the fast Fourier transform of the perturbed radial n = 1 magnetic field, caused by the rotation of the modes. The use of a carrier sinusoidal wave helps alleviating the spurious influence of non-sinusoidal magnetic perturbations induced by other instabilities like Edge localized modes (ELMs). The indicator constitutes a good candidate for further studies including machine learning approaches for mitigation and avoidance since, by deploying it systematically to evaluate the time instance for the expected locking, multi-machine databases can be populated. Furthermore, it can be thought as a contribution to a wider approach to dynamically tracking the chain of events leading to disruptions.

scholarly journals Information Gain in Event Space Reflects Chance and Necessity Components of an Event

Information ◽  
2019 ◽  
Vol 10 (11) ◽  
pp. 358 ◽  
Author(s):  
Georg F. Weber
Keyword(s):  
Phase Space ◽  
Ergodic Theorem ◽  
Information Gain ◽  
Characteristic Exponent ◽  
Vector Spaces ◽  
Event Space ◽  
Lyapunov Characteristic Exponent ◽  
Multiplicative Ergodic Theorem ◽  
Non Linear Systems ◽  
Eigenvectors And Eigenvalues

Information flow for occurrences in phase space can be assessed through the application of the Lyapunov characteristic exponent (multiplicative ergodic theorem), which is positive for non-linear systems that act as information sources and is negative for events that constitute information sinks. Attempts to unify the reversible descriptions of dynamics with the irreversible descriptions of thermodynamics have replaced phase space models with event space models. The introduction of operators for time and entropy in lieu of traditional trajectories has consequently limited—to eigenvectors and eigenvalues—the extent of knowable details about systems governed by such depictions. In this setting, a modified Lyapunov characteristic exponent for vector spaces can be used as a descriptor for the evolution of information, which is reflective of the associated extent of undetermined features. This novel application of the multiplicative ergodic theorem leads directly to the formulation of a dimension that is a measure for the information gain attributable to the occurrence. Thus, it provides a readout for the magnitudes of chance and necessity that contribute to an event. Related algorithms express a unification of information content, degree of randomness, and complexity (fractal dimension) in event space.

scholarly journals A New Dynamical Parameter for the Study of Sticky Orbits in a 3D Galactic Model

2011 ◽  
Vol 20 (3) ◽  
Author(s):  
Euaggelos E. Zotos
Keyword(s):  
Dynamical System ◽  
Classical Method ◽  
Elliptical Galaxy ◽  
Dynamical Model ◽  
Dynamical Parameter ◽  
2D System ◽  
Galactic Model ◽  
Dense Nucleus

AbstractA 3D dynamical model is used to study the motion in the central parts of an elliptical galaxy, hosting a massive and dense nucleus. Our aim is to investigate the regular or chaotic character of the motion, with emphasis in the different chaotic components, as well as the sticky regions of the dynamical system. In order to define the character of the motion in the 2D system, we use the classical method of the Poincaré x − p

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