Fluctuations In Initial Conditions Research Articles

Bifurcations, chaotic behavior, sensitivity analysis, and various soliton solutions for the extended nonlinear Schrödinger equation

In this manuscript, our primary objective is to delve into the intricacies of an extended nonlinear Schrödinger equation. To achieve this, we commence by deriving a dynamical system tightly linked to the equation through the Galilean transformation. We then employ principles from planar dynamical systems theory to explore the bifurcation phenomena exhibited within this derived system. To investigate the potential presence of chaotic behaviors, we introduce a perturbed term into the dynamical system and systematically analyze the extended nonlinear Schrödinger equation. This investigation is further enriched by the presentation of comprehensive two- and 3D phase portraits. Moreover, we conduct a meticulous sensitivity analysis of the dynamical system using the Runge–Kutta method. Through this analytical process, we confirm that minor fluctuations in initial conditions have only minimal effects on solution stability. Additionally, we utilize the complete discrimination system of the polynomial method to systematically construct single traveling wave solutions for the governing model.

Isolating Patterns in Open Reaction\u2013Diffusion Systems

Realistic examples of reaction–diffusion phenomena governing spatial and spatiotemporal pattern formation are rarely isolated systems, either chemically or thermodynamically. However, even formulations of ‘open’ reaction–diffusion systems often neglect the role of domain boundaries. Most idealizations of closed reaction–diffusion systems employ no-flux boundary conditions, and often patterns will form up to, or along, these boundaries. Motivated by boundaries of patterning fields related to the emergence of spatial form in embryonic development, we propose a set of mixed boundary conditions for a two-species reaction–diffusion system which forms inhomogeneous solutions away from the boundary of the domain for a variety of different reaction kinetics, with a prescribed uniform state near the boundary. We show that these boundary conditions can be derived from a larger heterogeneous field, indicating that these conditions can arise naturally if cell signalling or other properties of the medium vary in space. We explain the basic mechanisms behind this pattern localization and demonstrate that it can capture a large range of localized patterning in one, two, and three dimensions and that this framework can be applied to systems involving more than two species. Furthermore, the boundary conditions proposed lead to more symmetrical patterns on the interior of the domain and plausibly capture more realistic boundaries in developmental systems. Finally, we show that these isolated patterns are more robust to fluctuations in initial conditions and that they allow intriguing possibilities of pattern selection via geometry, distinct from known selection mechanisms.

Study of the influence of initial-state fluctuations on hydrodynamic simulations

In this work, we are focusing on assessing the contribution of the initial-state fluctuations of heavy ion collision in the hydrodynamic simulations. We are trying to answer the question of whether the hydrodynamic simulation retains the same level of fluctuation in the final-state as for the initial stage. In another scenario, the hydrodynamic simulations of the fluctuation drowns in the final distribution of expanding matter. For this purpose, we prepared sufficient relativistic hydrodynamic program to study A+A interaction which allows analysing initial-state fluctuations in the bulk nuclear matter. For such an assumption, it is better to use high spatial resolution. Therefore, we applied the (3+1) dimensional Cartesian coordinate system. We implemented our program using parallel computing on graphics cards processors - Graphics Processing Unit (GPU). Simulations were carried out with various levels of fluctuation in initial conditions using the average method of events coming from UrQMD models. Energy density distributions were analysed and the contribution of fluctuations in initial conditions was assessed in the hydrodynamic simulation.

E-by-e fluctuations of initial conditions in interactions of light, intermediate and heavy nuclei at energies 4 - 200 GeV per nucleon

A better understanding of initial conditions of nuclear interactions is one of the most important heavy ion physics problems. The paper considers these problems in the centrality selected interactions of light (C, O, Ne) – (beam) and heavy (Au, Pb) – (beam) ions with light (C/N/O) – (target) and heavy (Ag/Br) – (target) nuclei using the data of JINR-AGS-SPS target emulsion experiments with limited statistics. To reveal a possible system structure dependence, the analyses of E-by-E fluctuations on the longitudinal event shape and multiplicity have been performed, vertically and horizontally.

Measurement-enhanced determination of BEC phase transitions

We demonstrate how dispersive atom number measurements during evaporative cooling can be used for enhanced determination of the parameter dependence of the transition to a Bose–Einstein condensate (BEC). In this way shot-to-shot fluctuations in initial conditions are detected and the information extracted per experimental realization is increased. We furthermore calibrate in situ images from dispersive probing of a BEC with corresponding absorption images in time-of-flight. This allows for the determination of the transition point in a single experimental realization by applying multiple dispersive measurements. Finally, we explore the continuous probing of several consecutive phase transition crossings using the periodic addition of a focused ‘dimple’ potential.

Weak fault monitoring method for batch process based on multi-model SDKPCA

In industrial manufacturing, most batch processes have the dynamic and nonlinear features in nature. To ensure both quality consistency of the manufactured products and safe operation of this kind of batch process, a number of multivariate statistical analyses, including multiway principal component analysis (MPCA), batch dynamic kernel principal component analysis (BDKPCA), have been developed in recent years. However, these methods can't effectively detect the weak faults due to large fluctuations in the initial conditions, because the weak faults are submerged to the fluctuations in the poor initial conditions. In order to improve the performance of the weak fault detection, a new nonlinear dynamic batch process monitoring method, called multi-model single dynamic kernel principal component analysis (M-SDKPCA), is proposed in this paper. The multi-model methodology is based on BDKPCA. The method firstly integrates kernel PCA (KPCA) and auto-regressive moving average exogenous (ARMAX) time series model for each batch data at each stage to build SDKPCA. Then hierarchical clusters are obtained through load matrix similarity among SDKPCA models. At different stages, multiple model structures are constructed along with the variation of the cluster number. The monitoring method proposed in this paper was applied to fault detection for benchmark of fed-batch penicillin production. In both off-line analysis and on-line batch monitoring, the proposed approach shows better performance than MKPCA and BDKPCA.

Influence of stochastic domain growth on pattern nucleation for diffusive systems with internal noise

Numerous mathematical models exploring the emergence of complexity within developmental biology incorporate diffusion as the dominant mechanism of transport. However, self-organizing paradigms can exhibit the biologically undesirable property of extensive sensitivity, as illustrated by the behavior of the French-flag model in response to intrinsic noise and Turing's model when subjected to fluctuations in initial conditions. Domain growth is known to be a stabilizing factor for the latter, though the interaction of intrinsic noise and domain growth is underexplored, even in the simplest of biophysical settings. Previously, we developed analytical Fourier methods and a description of domain growth that allowed us to characterize the effects of deterministic domain growth on stochastically diffusing systems. In this paper we extend our analysis to encompass stochastically growing domains. This form of growth can be used only to link the meso- and macroscopic domains as the "box-splitting" form of growth on the microscopic scale has an ill-defined thermodynamic limit. The extension is achieved by allowing the simulated particles to undergo random walks on a discretized domain, while stochastically controlling the length of each discretized compartment. Due to the dependence of diffusion on the domain discretization, we find that the description of diffusion cannot be uniquely derived. We apply these analytical methods to two justified descriptions, where it is shown that, under certain conditions, diffusion is able to support a consistent inhomogeneous state that is far removed from the deterministic equilibrium, without additional kinetics. Finally, a logistically growing domain is considered. Not only does this show that we can deal with nonmonotonic descriptions of stochastic growth, but it is also seen that diffusion on a stationary domain produces different effects to diffusion on a domain that is stationary "on average."

Chaos in Bianchi I cosmology

We have investigated the dynamical evolution of the Yang–Mills (YM) field in Bianchi I cosmology background. We find that the long-time evolution behaviour of the YM field is highly sensitive to initial conditions, i.e. small fluctuations of initial conditions for fixed Hamiltonian of the system may rapidly change the evolution of the field. By using the Poincare section method, we further illustrate that the dynamical evolution of the YM field in Bianchi I cosmology background has certain typically chaotic properties.