Introduction

2020 ◽  
pp. 3-16
Author(s):  
Kaloshin Vadim ◽  
Zhang Ke
Keyword(s):  
Degrees Of Freedom ◽  
Large Scale ◽  
Energy Surface ◽  
Initial Conditions ◽  
Invariant Tori ◽  
Kam Theory ◽  
Ergodic Hypothesis ◽  
Open Set ◽  
Nearly Integrable Systems

This introductory chapter provides an overview of Arnold diffusion. The famous question called the ergodic hypothesis, formulated by Maxwell and Boltzmann, suggests that for a typical Hamiltonian on a typical energy surface, all but a set of initial conditions of zero measure have trajectories dense in this energy surface. However, Kolmogorov-Arnold-Moser (KAM) theory showed that for an open set of (nearly integrable) Hamiltonian systems, there is a set of initial conditions of positive measure with almost periodic trajectories. This disproved the ergodic hypothesis and forced reconsideration of the problem. For autonomous nearly integrable systems of two degrees or time-periodic systems of one and a half degrees of freedom, the KAM invariant tori divide the phase space. These invariant tori forbid large scale instability. When the degrees of freedoms are larger than two, large scale instability is indeed possible, as evidenced by the examples given by Vladimir Arnold. The chapter explains that the book answers the question of the typicality of these instabilities in the two and a half degrees of freedom case.

ANALYTICAL EXPRESSIONS FOR STABLE AND UNSTABLE MANIFOLDS IN HIGHER DEGREE OF FREEDOM HAMILTONIAN SYSTEMS

1996 ◽  
Vol 06 (11) ◽  
pp. 1997-2013 ◽  
Author(s):  
HARRY DANKOWICZ
Keyword(s):  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Initial Conditions ◽  
Saddle Points ◽  
Melnikov Method ◽  
Type Systems ◽  
Stable And Unstable Manifolds ◽  
Unstable Manifolds ◽  
Analytical Expressions

Perturbations of completely integrable Hamiltonian systems with three or more degrees of freedom are studied. In particular, the unperturbed systems are assumed to be separable into a product of simple oscillator-type systems and a system containing homo- or heteroclinic connections consisting of stable and unstable manifolds of saddle points. Under a perturbation, the manifolds persist but separate and may no longer intersect. In this paper we show how, with proper choices for initial conditions, one may solve the variational equations to obtain analytical expressions for orbits on the perturbed manifolds in the form of expansions in the small parameter characterizing the perturbation. The derivation also shows how the distance between the manifolds can be uniquely defined, and thus provides an alternative to the traditional higher dimensional Melnikov method. It is finally argued that the approximate knowledge of the shape and position of the perturbed manifolds could be utilized for the study of large-scale phase-space motions, such as those associated with Arnold diffusion. The theory is further illuminated in two example problems.

scholarly journals Complex Dynamics and Statistics of 1-D Hamiltonian Lattices: Long Range Interactions and Supratransmission

2020 ◽  
Vol 23 (2) ◽  
pp. 133-148
Author(s):  
Anastasios Bountis
Keyword(s):  
Degrees Of Freedom ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Invariant Tori ◽  
Kam Theory ◽  
Small Perturbations ◽  
Physical Systems ◽  
Long Range Interactions ◽  
The 1960S ◽  
Ordered Motion

In this paper, I review a number of results that my co-workers and I have obtained in the field of 1-Dimensional (1D) Hamiltonian lattices. This field has grown in recent years, due to its importance in revealing many phenomena that concern the occurrence of chaotic behavior in conservative physical systems with a high number of degrees of freedom. After the establishment of the Kolomogorov-Arnol'd-Moser (KAM) theory in the 1960s, a wealth of results were obtained about such systems as small perturbations of completely integrable Ndegree- of-freedom Hamiltonians, where ordered motion is dominant in the form of invariant tori. Since the 1980s, however, and particularly in the last two decades, there has been great progress in understanding the properties of Hamiltonian 1D lattices far from the KAM regime, where "weak" and "strong" forms of chaos begin to play an increasingly significant role. It is the purpose of this review to address and highlight some of these advances, in which the author has made several contributions concerning the dynamics and statistics of these lattices.

scholarly journals Features and new physical scales in primordial observables: Theory and observation

2015 ◽  
Vol 24 (10) ◽  
pp. 1530023 ◽  
Author(s):  
Jens Chluba ◽  
Jan Hamann ◽  
Subodh P. Patil
Keyword(s):  
Correlation Functions ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Initial Conditions ◽  
Linear Response Theory ◽  
Microwave Background ◽  
Scale Invariant ◽  
Cosmological Observations ◽  
Background Models ◽  
Characteristic Scales

All cosmological observations to date are consistent with adiabatic, Gaussian and nearly scale invariant initial conditions. These findings provide strong evidence for a particular symmetry breaking pattern in the very early universe (with a close to vanishing order parameter, ϵ), widely accepted as conforming to the predictions of the simplest realizations of the inflationary paradigm. However, given that our observations are only privy to perturbations, in inferring something about the background that gave rise to them, it should be clear that many different underlying constructions project onto the same set of cosmological observables. Features in the primordial correlation functions, if present, would offer a unique and discriminating window onto the parent theory in which the mechanism that generated the initial conditions is embedded. In certain contexts, simple linear response theory allows us to infer new characteristic scales from the presence of features that can break the aforementioned degeneracies among different background models, and in some cases can even offer a limited spectroscopy of the heavier degrees of freedom that couple to the inflaton. In this review, we offer a pedagogical survey of the diverse, theoretically well-grounded mechanisms which can imprint features into primordial correlation functions in addition to reviewing the techniques one can employ to probe observations. These observations include cosmic microwave background (CMB) anisotropies and spectral distortions as well as the matter two and three point functions as inferred from large-scale structure (LSS) and potentially, 21 cm surveys.

Stochastic metastability and Hamiltonian dynamics

1979 ◽  
Vol 366 (1724) ◽  
pp. 129-141 ◽  
Keyword(s):  
Invariant Torus ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Hamiltonian Dynamics ◽  
Invariant Tori ◽  
Density Theorem ◽  
Maximum Displacement ◽  
Physical Systems ◽  
Definition Of ◽  
Almost All

Stochastic metastability is stability weakened by chance. In a definition of stability certain conditions must be satisfied for sufficiently small displacements. Corresponding to a definition of stability we introduce a definition of (stochastic) metastability, in which the same conditions must be satisfied with probability approaching unity as the maximum displacement approaches zero. Measurable sets are essential to metastability as open sets are to stability. Because of Arnol’d diffusion, the invariant tori of conservative Hamiltonian systems of n ≽ 3 degrees of freedom are not usually dynamically stable with respect to changes in initial conditions. However, we prove using the Lebesgue density theorem that if the proper invariant tori of phase space fill a bounded domain of positive measure, then they are almost all dynamically metastable. Metastability is difficult to distinguish from stability in physical systems or numerical experiments, so the result is consistent with the difficulty of observing Arnol’d diffusion in the neighbourhood of an invariant torus in practice. Probability and measure are essential to the general theory of stability of nonlinear systems.

scholarly journals The anisotropy of the power spectrum in periodic cosmological simulations

2021 ◽  
Vol 503 (4) ◽  
pp. 5638-5645
Author(s):  
Gábor Rácz ◽  
István Szapudi ◽  
István Csabai ◽  
László Dobos
Keyword(s):  
Dark Matter ◽  
Power Spectrum ◽  
Large Scale ◽  
Cold Dark Matter ◽  
Initial Conditions ◽  
Power Spectra ◽  
Cosmological Simulations ◽  
Spherical Collapse ◽  
Lower Amplitude ◽  
Hydrodynamical Simulations

ABSTRACT The classical gravitational force on a torus is anisotropic and always lower than Newton’s 1/r2 law. We demonstrate the effects of periodicity in dark matter only N-body simulations of spherical collapse and standard Lambda cold dark matter (ΛCDM) initial conditions. Periodic boundary conditions cause an overall negative and anisotropic bias in cosmological simulations of cosmic structure formation. The lower amplitude of power spectra of small periodic simulations is a consequence of the missing large-scale modes and the equally important smaller periodic forces. The effect is most significant when the largest mildly non-linear scales are comparable to the linear size of the simulation box, as often is the case for high-resolution hydrodynamical simulations. Spherical collapse morphs into a shape similar to an octahedron. The anisotropic growth distorts the large-scale ΛCDM dark matter structures. We introduce the direction-dependent power spectrum invariant under the octahedral group of the simulation volume and show that the results break spherical symmetry.

scholarly journals Dynamic Loads and Response of a Spar Buoy Wind Turbine with Pitch-Controlled Rotating Blades: An Experimental Study

Energies ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 3598
Author(s):  
Sara Russo ◽  
Pasquale Contestabile ◽  
Andrea Bardazzi ◽  
Elisa Leone ◽  
Gregorio Iglesias ◽  
...  
Keyword(s):  
Wind Speed ◽  
Wind Turbine ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Laboratory Data ◽  
Nonlinear Behavior ◽  
Offshore Wind ◽  
Small Scale ◽  
Wave Steepness ◽  
Design Factor

New large-scale laboratory data are presented on a physical model of a spar buoy wind turbine with angular motion of control surfaces implemented (pitch control). The peculiarity of this type of rotating blade represents an essential aspect when studying floating offshore wind structures. Experiments were designed specifically to compare different operational environmental conditions in terms of wave steepness and wind speed. Results discussed here were derived from an analysis of only a part of the whole dataset. Consistent with recent small-scale experiments, data clearly show that the waves contributed to most of the model motions and mooring loads. A significant nonlinear behavior for sway, roll and yaw has been detected, whereas an increase in the wave period makes the wind speed less influential for surge, heave and pitch. In general, as the steepness increases, the oscillations decrease. However, higher wind speed does not mean greater platform motions. Data also indicate a significant role of the blade rotation in the turbine thrust, nacelle dynamic forces and power in six degrees of freedom. Certain pairs of wind speed-wave steepness are particularly unfavorable, since the first harmonic of the rotor (coupled to the first wave harmonic) causes the thrust force to be larger than that in more energetic sea states. The experiments suggest that the inclusion of pitch-controlled, variable-speed blades in physical (and numerical) tests on such types of structures is crucial, highlighting the importance of pitch motion as an important design factor.

scholarly journals EDGE: a new approach to suppressing numerical diffusion in adaptive mesh simulations of galaxy formation

2020 ◽  
Vol 501 (2) ◽  
pp. 1755-1765
Author(s):  
Andrew Pontzen ◽  
Martin P Rey ◽  
Corentin Cadiou ◽  
Oscar Agertz ◽  
Romain Teyssier ◽  
...  
Keyword(s):  
Star Formation ◽  
Galaxy Formation ◽  
Adaptive Mesh Refinement ◽  
Large Scale ◽  
Initial Conditions ◽  
Dwarf Galaxy ◽  
High Redshift ◽  
Morphological Structure ◽  
Adaptive Mesh ◽  
Numerical Diffusion

ABSTRACT We introduce a new method to mitigate numerical diffusion in adaptive mesh refinement (AMR) simulations of cosmological galaxy formation, and study its impact on a simulated dwarf galaxy as part of the ‘EDGE’ project. The target galaxy has a maximum circular velocity of $21\, \mathrm{km}\, \mathrm{s}^{-1}$ but evolves in a region that is moving at up to $90\, \mathrm{km}\, \mathrm{s}^{-1}$ relative to the hydrodynamic grid. In the absence of any mitigation, diffusion softens the filaments feeding our galaxy. As a result, gas is unphysically held in the circumgalactic medium around the galaxy for $320\, \mathrm{Myr}$, delaying the onset of star formation until cooling and collapse eventually triggers an initial starburst at z = 9. Using genetic modification, we produce ‘velocity-zeroed’ initial conditions in which the grid-relative streaming is strongly suppressed; by design, the change does not significantly modify the large-scale structure or dark matter accretion history. The resulting simulation recovers a more physical, gradual onset of star formation starting at z = 17. While the final stellar masses are nearly consistent ($4.8 \times 10^6\, \mathrm{M}_{\odot }$ and $4.4\times 10^6\, \mathrm{M}_{\odot }$ for unmodified and velocity-zeroed, respectively), the dynamical and morphological structure of the z = 0 dwarf galaxies are markedly different due to the contrasting histories. Our approach to diffusion suppression is suitable for any AMR zoom cosmological galaxy formation simulations, and is especially recommended for those of small galaxies at high redshift.

scholarly journals Automated Quality Assessment of Interferometric Ring Laser Data

Sensors ◽  
2021 ◽  
Vol 21 (10) ◽  
pp. 3425
Author(s):  
Andreas Brotzer ◽  
Felix Bernauer ◽  
Karl Ulrich Schreiber ◽  
Joachim Wassermann ◽  
Heiner Igel
Keyword(s):  
Quality Assessment ◽  
Ground Motion ◽  
Ring Laser ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Scientific Field ◽  
Optical Frequency ◽  
Laser Array ◽  
Laser Data ◽  
Rotational Motions

In seismology, an increased effort to observe all 12 degrees of freedom of seismic ground motion by complementing translational ground motion observations with measurements of strain and rotational motions could be witnessed in recent decades, aiming at an enhanced probing and understanding of Earth and other planetary bodies. The evolution of optical instrumentation, in particular large-scale ring laser installations, such as G-ring and ROMY (ROtational Motion in seismologY), and their geoscientific application have contributed significantly to the emergence of this scientific field. The currently most advanced, large-scale ring laser array is ROMY, which is unprecedented in scale and design. As a heterolithic structure, ROMY’s ring laser components are subject to optical frequency drifts. Such Sagnac interferometers require new considerations and approaches concerning data acquisition, processing and quality assessment, compared to conventional, mechanical instrumentation. We present an automated approach to assess the data quality and the performance of a ring laser, based on characteristics of the interferometric Sagnac signal. The developed scheme is applied to ROMY data to detect compromised operation states and assign quality flags. When ROMY’s database becomes publicly accessible, this assessment will be employed to provide a quality control feature for data requests.

scholarly journals Relativistic Cosmology with an Introduction to Inflation

Universe ◽  
2021 ◽  
Vol 7 (8) ◽  
pp. 276
Author(s):  
Muhammad Zahid Mughal ◽  
Iftikhar Ahmad ◽  
Juan Luis García Guirao
Keyword(s):  
Standard Model ◽  
Early Universe ◽  
Large Scale ◽  
Initial Conditions ◽  
Big Bang ◽  
Relativistic Cosmology ◽  
The Standard Model ◽  
Big Bang Model ◽  
The Big Bang ◽  
The Universe

In this review article, the study of the development of relativistic cosmology and the introduction of inflation in it as an exponentially expanding early phase of the universe is carried out. We study the properties of the standard cosmological model developed in the framework of relativistic cosmology and the geometric structure of spacetime connected coherently with it. The geometric properties of space and spacetime ingrained into the standard model of cosmology are investigated in addition. The big bang model of the beginning of the universe is based on the standard model which succumbed to failure in explaining the flatness and the large-scale homogeneity of the universe as demonstrated by observational evidence. These cosmological problems were resolved by introducing a brief acceleratedly expanding phase in the very early universe known as inflation. The cosmic inflation by setting the initial conditions of the standard big bang model resolves these problems of the theory. We discuss how the inflationary paradigm solves these problems by proposing the fast expansion period in the early universe. Further inflation and dark energy in fR modified gravity are also reviewed.

scholarly journals Tropical Cyclone Count Forecasting Using a Dynamical Seasonal Prediction System: Sensitivity to Improved Ocean Initialization

2011 ◽  
Vol 24 (12) ◽  
pp. 2963-2982 ◽  
Author(s):  
Andrea Alessandri ◽  
Andrea Borrelli ◽  
Silvio Gualdi ◽  
Enrico Scoccimarro ◽  
Simona Masina
Keyword(s):  
Tropical Cyclone ◽  
Indian Ocean ◽  
Wind Shear ◽  
Large Scale ◽  
Initial Conditions ◽  
North Pacific Ocean ◽  
Convective Available Potential Energy ◽  
Seasonal Prediction ◽  
Boreal Summer ◽  
Prediction System

Abstract This study investigates the predictability of tropical cyclone (TC) seasonal count anomalies using the Centro Euro-Mediterraneo per i Cambiamenti Climatici–Istituto Nazionale di Geofisica e Vulcanologia (CMCC-INGV) Seasonal Prediction System (SPS). To this aim, nine-member ensemble forecasts for the period 1992–2001 for two starting dates per year were performed. The skill in reproducing the observed TC counts has been evaluated after the application of a TC location and tracking detection method to the retrospective forecasts. The SPS displays good skill in predicting the observed TC count anomalies, particularly over the tropical Pacific and Atlantic Oceans. The simulated TC activity exhibits realistic geographical distribution and interannual variability, thus indicating that the model is able to reproduce the major basic mechanisms that link the TCs’ occurrence with the large-scale circulation. TC count anomalies prediction has been found to be sensitive to the subsurface assimilation in the ocean for initialization. Comparing the results with control simulations performed without assimilated initial conditions, the results indicate that the assimilation significantly improves the prediction of the TC count anomalies over the eastern North Pacific Ocean (ENP) and northern Indian Ocean (NI) during boreal summer. During the austral counterpart, significant progresses over the area surrounding Australia (AUS) and in terms of the probabilistic quality of the predictions also over the southern Indian Ocean (SI) were evidenced. The analysis shows that the improvement in the prediction of anomalous TC counts follows the enhancement in forecasting daily anomalies in sea surface temperature due to subsurface ocean initialization. Furthermore, the skill changes appear to be in part related to forecast differences in convective available potential energy (CAPE) over the ENP and the North Atlantic Ocean (ATL), in wind shear over the NI, and in both CAPE and wind shear over the SI.

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