A new characterization of nonisotropic chaotic vibrations of the one-dimensional linear wave equation with a van der Pol boundary condition

AbstractThe one-dimensional linear wave equation with a van der Pol nonlinear boundary condition is one of the simplest models that may cause isotropic or nonisotropic chaotic vibrations (Trans. Amer. Math. Soc. 350 (1998) 4265–4311, Internat. J. Bifur. Chaos 8 (1998) 423–445, Internat. J. Bifur. Chaos 8 (1998) 447–470, J. Math. Phys. 39 (1998) 6459–6489, Internat. J. Bifur. Chaos 12 (2002) 535–559). In this paper, we characterize nonisotropic chaotic vibration by means of the total variation theory. We obtain the classification results on the growth of the total variation of the snapshots on the spatial interval in the long-time horizon with respect to two parameters entering different regimes in R2

Study of Nonlinear Analysis and Chaos in Vibrations and Fluids

Chaos and turbulence are two important topics in nonlinear dynamics. In this study, two problems related to chaos and turbulence modelling are presented. They are the chaotic vibration phenomenon in high-dimensional partial differential equations and the emergence of the Navier-Stokes-alpha model for channel flows. The study of the chaotic vibration phenomenon in high-dimensional partial differential equations is explained from both the numerical and theoretical aspects. In the numerical perspective, we have studied the chaotic vibration phenomenon of the 2D wave equation through numerical simulations. Based on the finite-volume method, we have built our own solver “img2Foam" in the Computational Fluid Dynamics software OpenFOAM (Open source Field Operation and Manipulation). We have implemented several numerical simulations containing both chaotic and non-chaotic cases. As for the theoretical perspective, we give a rigorous proof for the chaotic vibration phenomenon of the 2D non-strictly hyperbolic equation. After introducing two linear operators, the initial system of the 2D non-strictly hyperbolic equation is converted into a system of two coupled first order equations. By using the method of characteristics, we have found the explicit solution formulas of the new system. We have also found a regime of the parameters when the chaotic vibration phenomenon occurs by applying the period-doubling bifurcation theorem. Numerical simulations are presented to validate the theoretical results. Inspired by the concept of the regular part of the weak attractor of the 3D Navier- Stokes equations, we concentrate on a restricted class of fluid flows to explore the transition from the Navier-Stokes equations to the Navier-Stokes-alpha model for channel flows. The Navier-Stokes equations have been widely used to describe the motion of viscous incompressible fluid flows. As an averaged version of the Navier- Stokes equations, the Navier-Stokes-alpha model has solid mathematical properties as well as reliable experimental matches. Therefore, the Navier-Stokes-alpha model is taken as an approximation for the dynamics of appropriately averaged turbulent fluid flows. We are interested in finding a connection between Navier-Stokes equations and the Navier-Stokes-alpha model in terms of the physical properties of the fluid flow. Given the hypothesis that the turbulence described by the Navier-Stokes-alpha model was partly due to the roughness of the walls, the transition from the Navier-Stokes equations into the Navier-Stokes-alpha model is presented by introducing a Reynolds type averaging

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Spacelab Science Results Study

Beginning with OSTA-1 in November 1981 and ending with Neurolab in March 1998, a total of 36 Shuttle missions carried various Spacelab components such as the Spacelab module, pallet, instrument pointing system, or mission peculiar experiment support structure. The experiments carried out during these flights included astrophysics, solar physics, plasma physics, atmospheric science, Earth observations, and a wide range of microgravity experiments in life sciences, biotechnology, materials science, and fluid physics which includes combustion and critical point phenomena. In all, some 764 experiments were conducted by investigators from the U.S., Europe, and Japan. The purpose of this Spacelab Science Results Study is to document the contributions made in each of the major research areas by giving a brief synopsis of the more significant experiments and an extensive list of the publications that were produced. We have also endeavored to show how these results impacted the existing body of knowledge, where they have spawned new fields, and if appropriate, where the knowledge they produced has been applied