scholarly journals ANALYSIS OF ROUND OFF ERRORS WITH REVERSIBILITY TEST AS A DYNAMICAL INDICATOR

2012 ◽  
Vol 22 (09) ◽  
pp. 1250215 ◽  
Author(s):  
DAVIDE FARANDA ◽  
MARTÍN FEDERICO MESTRE ◽  
GIORGIO TURCHETTI
Keyword(s):  
Phase Space ◽  
Initial Conditions ◽  
Characteristic Exponent ◽  
Random Perturbations ◽  
Symplectic Maps ◽  
Lyapunov Characteristic Exponent ◽  
The Mean ◽  
Dynamical Properties ◽  
Discrete Time Dynamical Systems ◽  
Number Of Iterations

We compare the divergence of orbits and the reversibility error for discrete time dynamical systems. These two quantities are used to explore the behavior of the global error induced by round off in the computation of orbits. The similarity of results found for any system we have analyzed suggests the use of the reversibility error, whose computation is straightforward since it does not require the knowledge of the exact orbit, as a dynamical indicator. The statistics of fluctuations induced by round off for an ensemble of initial conditions has been compared with the results obtained in the case of random perturbations. Significant differences are observed in the case of regular orbits due to the correlations of round off error, whereas the results obtained for the chaotic case are nearly the same. Both the reversibility error and the orbit divergence computed for the same number of iterations on the whole phase space provide an insight on the local dynamical properties with a detail comparable with other dynamical indicators based on variational methods such as the finite time maximum Lyapunov characteristic exponent, the mean exponential growth factor of nearby orbits and the smaller alignment index. For 2D symplectic maps, the differentiation between regular and chaotic regions is well full-filled. For 4D symplectic maps, the structure of the resonance web as well as the nearby weakly chaotic regions are accurately described.

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 Revealing the Character of Orbits in a Binary System Consisting of a Primary Galaxy and a Satellite Companion

2013 ◽  
Vol 30 ◽  
Author(s):  
Euaggelos E. Zotos
Keyword(s):  
Phase Space ◽  
Initial Conditions ◽  
Classical Method ◽  
Stellar System ◽  
Characteristic Exponent ◽  
Low Energy ◽  
Dynamical Analysis ◽  
The Galaxy ◽  
Chaotic Nature ◽  
2D And 3D

AbstractIn this article, we present a galactic gravitational model of three degrees of freedom (3D), in order to study and reveal the character of the orbits of the stars, in a binary stellar system composed of a primary quiet or active galaxy and a small satellite companion galaxy. Our main dynamical analysis will be focused on the behaviour of the primary galaxy. We investigate in detail the regular or chaotic nature of motion, in two different cases: (i) the time-independent model in both 2D and 3D dynamical systems and (ii) the time-evolving 3D model. For the description of the structure of the 2D system, we use the classical method of the Poincaré (x, px), y = 0, py < 0 phase plane. In order to study the structure of the phase space of the 3D system, we take sections in the plane y = 0 of the 3D orbits, whose initial conditions differ from the plane parent periodic orbits, only by the z component. The set of the four-dimensional points in the (x, px, z, pz) phase space is projected on the (z, pz) plane. The maximum Lyapunov characteristic exponent is used in order to make an estimation of the chaoticity of our galactic system, in both 2D and 3D dynamical models. Our numerical calculations indicate that the percentage of the chaotic orbits increases when the primary galaxy has a dense and massive nucleus. The presence of the dense galactic core also increases the stellar velocities near the center of the galaxy. Moreover, for small values of the distance R between the two bodies, low-energy stars display chaotic motion, near the central region of the galaxy, while for larger values of the distance R, the motion in active galaxies is entirely regular for low-energy stars. Our simulations suggest that in galaxies with a satellite companion, the chaotic nature of motion is not only a result of the galactic interaction between the primary galaxy and its companion, but also a result caused by the presence of the dense nucleus in the core of the primary galaxy. Theoretical arguments are presented in order to support and interpret the numerically derived outcomes. Furthermore, we follow the 3D evolution of the primary galaxy, when mass is transported adiabatically from the disk to the nucleus. Our numerical results are in satisfactory agreement with observational data obtained from the M51-type binary stellar systems. A comparison between the present research and similar and earlier work is also made.

scholarly journals On Asymmetric Periodic Solutions of the Plane Restricted Problem of Three Bodies, and Bifurcations of Families

1978 ◽  
Vol 41 ◽  
pp. 319-323
Author(s):  
P.J. Message ◽  
D.B. Taylor
Keyword(s):  
Periodic Solutions ◽  
Periodic Orbits ◽  
Restricted Problem ◽  
Initial Conditions ◽  
Characteristic Exponent ◽  
Mean Value ◽  
Mirror Image ◽  
Mean Values ◽  
Major Semi Axis ◽  
The Mean

Previous work on the plane circular restricted problem of three bodies (Message 1953, 1959, 1970, and Fragakis 1973) has shown the existence, in association with each of the commensurabilities 2:1 and 3:1 of the orbital periods, of a pair of families of asymmetric periodic solutions, branching from the stable series of symmetric periodic solutions of Poincaré’s second sort associated with that commensurability. (Each solution of either family is the mirror image, in the line of the two finite bodies, of a member of the other family of solutions associated with the commensurability.) The stability is transferred at the bifurcation to the two series of asymmetric orbits, each of which is therefore stable. Recent numerical integrations carried out by one of us (P.J.M.) have found such asymmetric periodic orbits associated also with the 4:1 commensurability, and quantities describing orbits of one of the two series are given in Table 1, showing the run of such orbits up to a second bifurcation with the same series of symmetric periodic orbits from which it sprang. Quantities describing some members of this series of symmetric orbits are given in Table 2. It is seen that stability is transferred back to the symmetric series at the second bifurcation. (The unit of distance is the distance between the two finite bodies, the unit of speed is the speed, of their relative motion, and the initial conditions given (x°, ẋ°, ẏ°) are for a crossing of the line of the two finite bodies, this line being taken as axis of “x” in a rotating Cartesian frame in the usual way. The mean values of the major semi-axis and eccentricity are denoted by ā and ē, respectively, C is Jacobi’s constant, and ȳ2 is the mean value of the critical argument ȳ2 = 4λ – λ′ – 3ω. The mass ratio used is 0.000954927, T is the period of the solution in units of the period of the motion of the two finite bodies, and 2π c/T is the non-zero characteristic exponent.)

Restricting Chaos to a Small Region in the Phase Space

1998 ◽  
Vol 08 (02) ◽  
pp. 401-407
Author(s):  
Zhihua Wu ◽  
Zhaoxuan Zhu ◽  
Chengfu Zhang
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Computer Simulations ◽  
Characteristic Exponent ◽  
Small Region ◽  
Dissipative Systems ◽  
Simple Method ◽  
Controlled System ◽  
Lyapunov Characteristic Exponent ◽  
Uncontrolled System

The idea of restricting chaos in dissipative systems to a small region in the phase space is proposed. The possibility of realization of this idea is demonstrated by applying a simple method summed up from computer simulations successfully to three different dynamical systems. It is found that not only does the trajectory of the controlled system occupy a region smaller than that of the uncontrolled chaotic system, the corresponding attractor of the Poincaré map is also smaller than that of the uncontrolled system. In addition, but also the maximum Lyapunov characteristic exponent of the system is greatly lowered.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

COLONY OF THE FUNGUS Aspergillus oryzae AND SELF-AFFINE FRACTAL GEOMETRY OF GROWTH FRONTS

Fractals ◽  
1993 ◽  
Vol 01 (01) ◽  
pp. 11-19 ◽  
Author(s):  
SHU MATSUURA ◽  
SASUKE MIYAZIMA
Keyword(s):  
Aspergillus Oryzae ◽  
Environmental Conditions ◽  
Fractal Geometry ◽  
Nutrient Concentration ◽  
Incubation Temperature ◽  
Characteristic Exponent ◽  
Growth Front ◽  
Favorable Condition ◽  
The Self ◽  
The Mean

A variety of colony shapes of the fungus Aspergillus oryzae under varying environmental conditions such as the nutrient concentration, medium stiffness and incubation temperature are obtained, ranging from a homogeneous Eden-like to a ramified DLA-like pattern. The roughness σ(l, h) of the growth front of the band-shaped colony, where h is the mean front height within l of the horizontal range, satisfies the self-affine fractal relation under favorable environmental conditions. In the most favorable condition of our experiments, its characteristic exponent is found to be a little larger than that of the 2-dimensional Eden model.

THE EFFECT OF PREMELTING: STUDY FOR SEMI-INFINITE CRYSTALS AND NANO-SYSTEMS

1994 ◽  
Vol 08 (24) ◽  
pp. 3411-3422 ◽  
Author(s):  
W. SCHOMMERS
Keyword(s):  
Pair Correlation ◽  
Mean Square Displacement ◽  
Phonon Density ◽  
Mean Square ◽  
Phonon Density Of States ◽  
Molecular Dynamics Calculations ◽  
Theory Of Melting ◽  
The Mean ◽  
Dynamical Properties ◽  
Reliable Interaction

The effect of premelting is of particular interest in connection with the theory of melting. In this paper, we discuss the structural and dynamical properties of the surfaces of semi-infinite crystals as well as of nano-clusters, which show the effect of premelting. The investigations are based on molecular-dynamics calculations: different models are used for the systematic study of the effect of premelting. In particular, the behaviour of the following functions have been studied: pair correlation function, generalized phonon density of states, and the mean-square displacement as a function of time. The calculations have been done for krypton since for this substance a reliable interaction potential is available.

scholarly journals Dynamical properties and extremes of Northern Hemisphere climate fields over the past 60 years

2017 ◽  
Vol 24 (4) ◽  
pp. 713-725 ◽  
Author(s):  
Davide Faranda ◽  
Gabriele Messori ◽  
M. Carmen Alvarez-Castro ◽  
Pascal Yiou
Keyword(s):  
Phase Space ◽  
Sea Level ◽  
Northern Hemisphere ◽  
Atmospheric Dynamics ◽  
Complex Nature ◽  
Human Welfare ◽  
Sea Level Pressure ◽  
Level Pressure ◽  
Dimensional Phase Space ◽  
Dynamical Properties

Abstract. Atmospheric dynamics are described by a set of partial differential equations yielding an infinite-dimensional phase space. However, the actual trajectories followed by the system appear to be constrained to a finite-dimensional phase space, i.e. a strange attractor. The dynamical properties of this attractor are difficult to determine due to the complex nature of atmospheric motions. A first step to simplify the problem is to focus on observables which affect – or are linked to phenomena which affect – human welfare and activities, such as sea-level pressure, 2 m temperature, and precipitation frequency. We make use of recent advances in dynamical systems theory to estimate two instantaneous dynamical properties of the above fields for the Northern Hemisphere: local dimension and persistence. We then use these metrics to characterize the seasonality of the different fields and their interplay. We further analyse the large-scale anomaly patterns corresponding to phase-space extremes – namely time steps at which the fields display extremes in their instantaneous dynamical properties. The analysis is based on the NCEP/NCAR reanalysis data, over the period 1948–2013. The results show that (i) despite the high dimensionality of atmospheric dynamics, the Northern Hemisphere sea-level pressure and temperature fields can on average be described by roughly 20 degrees of freedom; (ii) the precipitation field has a higher dimensionality; and (iii) the seasonal forcing modulates the variability of the dynamical indicators and affects the occurrence of phase-space extremes. We further identify a number of robust correlations between the dynamical properties of the different variables.

scholarly journals A modified fluctuation test for elucidating drug resistance in microbial and cancer cells

2020 ◽  
Author(s):  
Pavol Bokes ◽  
Abhyudai Singh
Keyword(s):  
Drug Resistance ◽  
Cancer Cells ◽  
Colony Size ◽  
Random Number ◽  
Initial Conditions ◽  
Fano Factor ◽  
Persister Cells ◽  
Efficient Treatment ◽  
Fluctuation Test ◽  
The Mean

AbstractClonal populations of microbial and cancer cells are often driven into a drug-tolerant persister state in response to drug therapy, and these persisters can subsequently adapt to the new drug environment via genetic and epigenetic mechanisms. Estimating the frequency with which drug-tolerance states arise, and its transition to drug-resistance, is critical for designing efficient treatment schedules. Here we study a stochastic model of cell proliferation where drug-tolerant persister cells transform into a drug-resistant state with a certain adaptation rate, and the resistant cells can then proliferate in the presence of the drug. Assuming a random number of persisters to begin with, we derive an exact analytical expression for the statistical moments and the distribution of the total cell count (i.e., colony size) over time. Interestingly, for Poisson initial conditions the noise in the colony size (as quantified by the Fano factor) becomes independent of the initial condition and only depends on the adaptation rate. Thus, experimentally quantifying the fluctuations in the colony sizes provides an estimate of the adaptation rate, which then can be used to infer the starting persister numbers from the mean colony size. Overall, our analysis introduces a modification of the classical Luria–Delbrück experiment, also called the “Fluctuation Test”, providing a valuable tool to quantify the emergence of drug resistance in cell populations.

scholarly journals Scaling laws and phase space analysis of a geomagnetic domino model

2021 ◽  
Vol 2090 (1) ◽  
pp. 012030
Author(s):  
K Peqini ◽  
D Prenga ◽  
R Osmanaj
Keyword(s):  
Phase Space ◽  
Scaling Laws ◽  
Outer Core ◽  
Main Field ◽  
Field Reversal ◽  
Core Dynamics ◽  
The Mean ◽  
Reversed Polarity ◽  
The Relationship ◽  
Mean Time

Abstract The geomagnetic field is among the most striking features of the Earth. By far the most important ingredient of it is generate in the fluid conductive outer core and it is known as the main field. It is characterized by a strong dipolar component as measured on the Earth’s surface. It is well established the fact that the dipolar component has reversed polarity many times, a phenomenon dubbed as dipolar field reversal (DFR). There have been proposed numerous models focused on describing the statistical features of the occurrence of such phenomena. One of them is the domino model, a simple toy model that despite its simplicity displays a very rich dynamic. This model incorporates several aspects of the outer core dynamics like the effect of rotation of Earth, the appearance of convective columns which create their own magnetic field, etc. In this paper we analyse the phase space of parameters of the model and identify several regimes. The two main regimes are the polarity changing one and the regime where the polarity remains the same. Also, we draw some scaling laws that characterize the relationship between the parameters and the mean time between reversals (mtr), the main output of the model.

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