Phase Space Of Initial Conditions Research Articles

Analytical solutions of ray-tracing equations in generalized elliptical anisotropy

Ray-tracing in anisotropic media is pivotal for interpreting observed seismic data and creating high-resolution images of subsurface structures, which are crucial in exploration geophysics. Elliptical anisotropy, a simplified model that approximates a transversely isotropic medium, is particularly relevant for geologic settings such as shale formations or stressed sedimentary layers where directional dependencies of seismic velocities are pronounced. We develop an analytical solution of the ray-tracing equations for a 2D inhomogeneous and anisotropic medium, where velocities depend elliptically on direction and increase linearly with depth — a scenario frequently encountered in stratified geologic formations. Unlike previous studies that assume constant ellipticity throughout the medium, our approach allows for variations in ellipticity, providing a more flexible and realistic representation of subsurface anisotropy. The phase velocities along the x- and z-axes are not necessarily multiples of each other at every point, offering a generalized version of the elliptical anisotropy. This enhancement may enable more accurate predictions and interpretations of observed seismic data, particularly in complex exploration scenarios. The analytical solution yields expressions for the ray paths and the wavefront normals. By setting the normals of the wavefront at the seismic source point and the location of the seismic source as the initial conditions in phase space, we explore the evolution of these wavefront normal curves across different types of the elliptical anisotropy. Our innovative approach includes plotting the evolution of wavefront normal curves on the generalized momentum coordinate plane of the phase space — that commonly overlooked in traditional models focused only on position coordinates.

Chaotic diffusion in the GJ 876 exoplanetary system

AbstractThis is a study about the dynamical stability of the GJ 876 exoplanetary system. The phase space of initial conditions (ICs) is characterized using the MEGNO indicator of chaos and the Shannon entropy approach for estimations of diffusivity. The results are compared to analyze correlations between the chaotic layers and the instability timescales. The long-term dynamical behavior of the system is reminiscent of the stable chaos, at least within the system’s lifetime.

An island of linearly stable non-hierarchical unequal mass periodic three-body orbits — A harbinger of circumbinary exoplanets?

We analyze all publicly available planar periodic three-body orbits in Newtonian gravity with zero angular momentum, which pass through a symmetrical collinear configuration. We find correlations between scale-invariant parameters, such as the ratio m3/m of the central and equal mass, the topological quantifier Lf of an orbit, the scale-invariant period T|E|3/2 , and the distribution of marginally linearly stable orbits. The elliptical (S) and marginal (M) linearly stable orbits are contained in a compact, non-zero measure domain with an apparently continuous boundary in the space spanned by the topological index Lf and the mass ratio m3/m. The distribution of S and M orbits predicts a sharp increase of both numbers of S and M orbits in this region as m3 is lowered below m/2. This indicates an island of stability of non-hierarchical three-body systems consisting of a light object (planet) orbiting two heavier ones (binary stars). We also find one-dimensional (sub)manifolds in the phase space of initial conditions that contain many of these linearly stable orbits. Orbits with increasingly longer periods populate/converge to these manifolds which suggests these manifolds are attractors.

A new generic and structurally stable cosmological model without singularity

Dynamical systems methods are used to investigate a cosmological model with non-minimally coupled scalar field and asymptotically quadratic potential function. We found that for values of the non-minimal coupling constant parameter 316<ξ<14 there exists an unstable asymptotic de Sitter state, free from a parallelly propagated singularity for 524≤ξ<14, giving rise to non-singular beginning of the universe. The energy density associated with this state depends on value of the non-minimal coupling constant and can be much smaller than the Planck energy density. For ξ=14 we found that the initial state is in form of the static Einstein universe. Proposed evolutional model, contrary to the seminal Starobinsky's model, do not depend on the specific choice of initial conditions in phase space, moreover, a small change in the model parameters do not change the evolution thus the model is generic and structurally stable. The values of the non-minimal coupling constant can indicate for a new fundamental symmetry in the gravitational theory. We show that Jordan frame and Einstein frame formulation of the theory are physically nonequivalent.

Wada index based on the weighted and truncated Shannon entropy

The Wada index based on the weighted and truncated Shannon entropy is presented in this paper. The proposed Wada index can detect if a given basin boundary is a Wada boundary. Moreover, the Wada index does represent the number and the distribution of different colors (attractors) in the two-dimensional phase space of initial conditions. The Wada index is based on the standard box counting algorithm, but computations in each box are based on the weighted and truncated Shannon entropy. That makes the algorithm for the computation of the Wada index conveniently applicable for different basins of attraction represented as color digital images. Computational experiments are used to demonstrate the advantages of the proposed index and compare it with other existing techniques and algorithms for the numerical analysis of Wada boundaries.

A test for fractal boundaries based on the basin entropy

In dynamical systems, basins of attraction connect a given set of initial conditions in phase space to their asymptotic states. The basin entropy and related tools quantify the unpredictability in the final state of a system when there is an initial perturbation or uncertainty in the initial state. Based on the basin entropy, the ln2 criterion allows for efficient testing of fractal basin boundaries at a fixed resolution. Here, we extend this criterion into a new test with improved sensitivity that we call the Sbbfractality test. Using the same single scale information, the Sbb fractality test allows for the detection of fractal boundaries in many more cases than the ln2 criterion. The new test is illustrated with the paradigmatic driven Duffing oscillator, and the results are compared with the classical approach given by the uncertainty exponent. We believe that this work can prove particularly useful to study both high-dimensional systems and experimental basins of attraction.

Rototaxis: Localization of active motion under rotation

The ability to navigate in complex, inhomogeneous environments is fundamental to survival at all length scales, giving rise to the rapid development of various subfields in bio-locomotion such as the well established concept of chemotaxis. In this work, we extend this existing notion of taxis to rotating environments and introduce the idea of roto-taxis to bio-locomotion. In particular, we explore both overdamped and inertial dynamics of a model synthetic self-propelled particle in the presence of constant global rotation, focusing on the particle's ability to localize near a rotation center as a survival strategy. We find that in the overdamped regime, the swimmer is in general able to generate a self restoring active torque that enables it to remain on stable epicyclical-like trajectories. On the other hand, for underdamped motion with inertial effects, the intricate competition between self-propulsion and inertial forces, in conjunction with the rototactic torque leads to complex dynamical behavior with non-trivial phase space of initial conditions which we reveal by numerical simulations. Our results are relevant for a wide range of setups, from vibrated granular matter on turntables to microorganisms or animals swimming near swirls or vortices.

Background Dynamics of Pre-inflationary Scenario in Brans-Dicke Loop Quantum Cosmology**Supported in part by the National Natural Science Foundation of China (NNSCF) under Grant Nos. 11375153 and 11675145. T.Z. is supported in part by NFSC under Grant No. 11675143 and the Fundamental Research Funds for the Provincial Universities of Zhejiang in China under Grant No. RF-A2019015

Recently the background independent nonperturbative quantization has been extended to various theories of gravity and the corresponding quantum effective cosmology has been derived, which provides us with necessary avenue to explore the pre-inflationary dynamics. Brans-Dicke (BD) loop quantum cosmology (LQC) is one of such theories whose effective background dynamics is considered in this article. Starting with a quantum bounce, we explore the pre-inflationary dynamics of a universe sourced by a scalar field with the Starobinsky potential in BD-LQC. Our study is based on the idea that though Einstein’s and Jordan’s frames are classically equivalent up to a conformal transformation in BD theory, this is no longer true after quantization. Taking the Jordan frame as the physical one we explore in detail the bouncing scenario which is followed by a phase of a slow roll inflation. The three phases of the evolution of the universe, namely, bouncing, transition from quantum bounce to classical universe, and the slow roll inflation, are noted for an initially kinetic energy dominated bounce. In addition, to be consistent with observations, we also identify the allowed phase space of initial conditions that would produce at least 60 e-folds of expansion during the slow roll inflation.

Scaling and self-similarity for the dynamics of a particle confined to an asymmetric time-dependent potential well

The dynamics of a classical point particle confined to an asymmetric time-dependent potential well is investigated under the framework of scaling. The potential corresponds to a reduced version of a particle moving along an infinitely periodic sequence of synchronously oscillating potential barriers. The dynamics of the model is described by a two-dimensional nonlinear and area preserving map in energy and phase variables. The asymmetric potential well is defined by two regions: Region I with fixed null potential and region II with an oscillating potential. The time-dependent potential of region II makes, for certain initial conditions, the particle to undergo a number of multiple reflections η at the border of the two regions and stay trapped in region I. Such trappings are described by histograms of multiple reflections η, obeying the power-law H(η)∝η^{-ν} with ν≈3, which are scale invariant with a scaling parameter depending of the control parameters of the mapping. We identify the location of the sets of initial conditions in phase space producing the multiple reflections and show that they generate well defined self-similar structures in density plots of trajectories in energy space. The self-similar structures can be enhanced by properly tuning the system parameters.

Vibrational predissociation dynamics of the nitric oxide dimer

We have studied the photofragmentation of NO dimer employing quasi-classical trajectories propagation using quantum probabilities distributions and Wigner distribution functions to determine the initial conditions of the system. The distributions of initial conditions in phase space, between Wigner distribution functions and quantum probabilities distributions, are not so different. The photofragmentation of NO dimer depends strongly on the excitation of the initial state, especially for the lowest of them. These results are in concordance with experimental values where we can see a similar tendency for the lifetimes different vibrational modes that have been studied. In this work we explore more intramolecular and intermolecular vibrational modes. We have increased the number of trajectories and the dissociation distance in the simulation as well.

Chirped resonance dynamics in phase space

The dynamics of passage and capture into resonance of a distribution of particles driven by a chirped frequency perturbation is discussed. The resonant capture in this case involves crossing of the separatrix by individual particles and, therefore, the adiabatic theorem cannot be used in studying this problem no matter how slow the variation of the driving frequency is. It is shown that, if instead of analysing complicated single particle dynamics in passage through resonance, one considers the slow evolution of a whole distribution of initial conditions in phase space, the adiabaticity and phase space incompressibility arguments yield a solution to the resonant passage problem. This approach is illustrated in the case of an ensemble of electrons driven by a chirped frequency wave passing through Cherenkov resonances with the velocity distribution of electrons.

Fingerprints of the initial conditions on the density profiles of cold and warm dark matter haloes

We use N-body simulations of dark matter haloes in cold dark matter (CDM) and a large set of different warm dark matter (WDM) cosmologies to demonstrate that the spherically averaged density profile of dark matter haloes has a shape that depends on the power spectrum of matter perturbations. Density profiles are steeper in WDM but become shallower at scales less than one percent of the virial radius. Virialization isotropizes the velocity dispersion in the inner regions of the halo but does not erase the memory of the initial conditions in phase space. The location of the observed deviations from CDM in the density profile and in phase space can be directly related to the ratio between the halo mass and the filtering mass and are most evident in small mass haloes, even for a 34 keV thermal relic WDM. The rearrangement of mass within the haloes supports analytic models of halo structure that include angular momentum. We also find evidence of a dependence of the slope of the inner density profile in CDM cosmologies on the halo mass with more massive haloes exhibiting steeper profiles, in agreement with the model predictions and with previous simulation results. Our work complements recent studies of microhaloes near the filtering scale in CDM and strongly argue against a universal shape for the density profile.

Effect of Nonlinear Dissipation on the Basin Boundaries of a Driven Two-Well Modified Rayleigh–Duffing Oscillator

This paper considers the effect of nonlinear dissipation on the basin boundaries of a driven two-well modified Rayleigh–Duffing oscillator where pure cubic, unpure cubic, pure quadratic and unpure quadratic nonlinearities are considered. By analyzing the potential, an analytic expression is found for the homoclinic orbit. The Melnikov criterion is used to examine a global homoclinic bifurcation and transition to chaos. Unpure quadratic parameter and parametric excitation amplitude effects are found on the critical Melnikov amplitude μ cr . Finally, the phase space of initial conditions is carefully examined in order to analyze the effect of the nonlinear damping, and particularly how the basin boundaries become fractalized.

Instability of quantum equilibrium in Bohm's dynamics.

We consider Bohm's second-order dynamics for arbitrary initial conditions in phase space. In principle, Bohm's dynamics allows for 'extended' non-equilibrium, with initial momenta not equal to the gradient of phase of the wave function (as well as initial positions whose distribution departs from the Born rule). We show that extended non-equilibrium does not relax in general and is in fact unstable. This is in sharp contrast with de Broglie's first-order dynamics, for which non-standard momenta are not allowed and which shows an efficient relaxation to the Born rule for positions. On this basis, we argue that, while de Broglie's dynamics is a tenable physical theory, Bohm's dynamics is not. In a world governed by Bohm's dynamics, there would be no reason to expect to see an effective quantum theory today (even approximately), in contradiction with observation.

On the longitudinal optimal perturbations to inviscid plane shear flow: formal solution and asymptotic approximation

AbstractWe study the longitudinal linear optimal perturbations (which maximize the energy gain up to a prescribed time $T$) to inviscid parallel shear flow, which present unbounded energy growth due to the lift-up mechanism. Using the phase invariance with respect to time, we show that for an arbitrary base flow profile and optimization time, the computation of the optimal longitudinal perturbation reduces to the resolution of a single one-dimensional eigenvalue problem valid for all times. The optimal perturbation and its amplification are then derived from the lowest eigenvalue and its associated eigenfunction, while the remainder of the infinite set of eigenfunctions provides an orthogonal base for decomposing the evolution of arbitrary perturbations. With this new formulation we obtain, asymptotically for large spanwise wavenumber ${k}_{z} , $ a prediction of the optimal gain and the localization of inviscid optimal perturbations for the two main classes of parallel flows: free shear flow with an inflectional velocity profile, and wall-bounded flow with maximum shear at the wall. We show that the inviscid optimal perturbations are localized around the point of maximum shear in a region with a width scaling like ${ k}_{z}^{- 1/ 2} $ for free shear flow, and like ${ k}_{z}^{- 2/ 3} $ for wall-bounded shear flows. This new derivation uses the stationarity of the base flow to transform the optimization of initial conditions in phase space into the optimization of a temporal phase along each trajectory, and an optimization among all trajectories labelled by their intersection with a codimension-1 subspace. The optimization of the time phase directly imposes that the initial and final energy growth rates of the optimal perturbation should be equal. This result requires only time invariance of the base flow, and is therefore valid for any linear optimal perturbation problem with stationary base flow.

The overshoot problem in inflation after tunneling

We show the absence of the usual parametrically large overshoot problem of small-field inflation if initiated by a Coleman-De Luccia (CDL) tunneling transition from an earlier vacuum in the limit of small inflationary scale compared to the tunneling scale. For low-power monomial exit potentials V(ϕ) ∼ ϕn , n < 4, we derive an expression for the amount of overshoot. This is bounded from above by the width of the steep barrier traversed after emerging from tunneling and before reaching a slow-roll region of the potential. For n ⩾ 4 we show that overshooting is entirely absent. We extend this result through binomials to a general potential written as a series expansion, and to the case of arbitrary finite initial speed of the inflaton. This places the phase space of initial conditions for small-field and large-field inflation on the same footing in a landscape of string theory vacua populated via CDL tunneling.

Time-dependent importance sampling in semiclassical initial value representation calculations for time correlation functions

An efficient time-dependent importance sampling method is developed for the Monte Carlo calculation of time correlation functions via the initial value representation (IVR) of semiclassical (SC) theory. A prefactor-free time-dependent sampling function weights the importance of a trajectory based on the magnitude of its contribution to the time correlation function, and global trial moves are used to facilitate the efficient sampling the phase space of initial conditions. The method can be generally applied to sampling rare events efficiently while avoiding being trapped in a local region of the phase space. Results presented in the paper for two system-bath models demonstrate the efficiency of this new importance sampling method for full SC-IVR calculations.

Brane inflation and the overshoot problem

We investigate recent claims that brane inflation solves the overshoot problem through a combination of microphysical restrictions on the phase space of initial conditions and the existence of the Dirac-Born-Infeld attractor in regimes where the slow-roll attractor does not apply. Carrying out a comprehensive analysis of the parameter space allowed by the latest advances in brane inflation model-building, we find that these restrictions are insufficient to solve the overshoot problem. The vast majority of the phase space of initial conditions is still dominated by overshoot trajectories. We present an analytic proof that the brane-inflationary attractor must be close to the slow-roll limit, and update the predictions for observables such as non-Gaussianity, cosmic string tension, and tensor modes.

Brane inflation is attractive

We study the phase space of initial conditions for brane inflation, and find that including the effects of the Dirac-Born-Infeld (DBI) kinetic term dramatically improves previous estimates on the amount of fine-tuning of initial conditions necessary for inflation, even for models dominated by slow roll. Two effects turn out to be important for the phase space analysis: restrictions on the total available phase space due to UV effects in brane inflation, and the extension of the inflationary attractor to the DBI inflationary regime. We compare the amount of initial conditions fine-tuning required for a brane-inflation model and its standard field theory counterpart and find that brane inflation decreases the required tuning by several orders of magnitude.

Effect of nonlinear dissipation on the basin boundaries of a driven two-well Rayleigh–Duffing oscillator

The Rayleigh oscillator is one canonical example of self-excited systems. However, simple generalizations of such systems, such as the Rayleigh–Duffing oscillator, have not received much attention. The presence of a cubic term makes the Rayleigh–Duffing oscillator a more complex and interesting case to analyze. In this work, we use analytical techniques such as the Melnikov theory, to obtain the threshold condition for the occurrence of Smale-horseshoe type chaos in the Rayleigh–Duffing oscillator. Moreover, we examine carefully the phase space of initial conditions in order to analyze the effect of the nonlinear damping, and in particular how the basin boundaries become fractalized.