Analysis of stability, verification and chaos with the Kreiss-Yström equations

نویسندگان

  • William D. Fullmer
  • Martin A. Lopez de Bertodano
  • Min Chen
  • Alejandro Clausse
چکیده

A system of two coupled PDEs originally proposed and studied by Kreiss and Yström (2002), which is dynamically similar to a one-dimensional two-fluid model of two-phase flow, is investigated here. It is demonstrated that in the limit of vanishing viscosity (i.e., neglecting second-order and higher derivatives), the system possesses complex eigenvalues and is therefore ill-posed. The regularized problem (i.e., including viscous second-order derivatives) retains the long-wavelength linear instability but with a cut-off wavelength, below which the system is linearly stable and dissipative. A second-order accurate numerical scheme, which is verified using the method of manufactured solutions, is used to simulate the system. For short to intermediate periods of time, numerical solutions compare favorably to those published previously by the original authors. However, the solutions at a later time are considerably different and have the properties of chaos. To quantify the chaos, the largest Lyapunov exponent is calculated and found to be approximately 0.38. Additionally, the correlation dimension of the attractor is assessed, resulting in a fractal dimension of 2.8 with an embedded dimension of approximately 6. Subsequently, the route to chaos is qualitatively explored with investigations of asymptotic stability, traveling-wave limit cycles and intermittency. Finally, the numerical solution, which is grid-dependent in space–time for long times, is demonstrated to be convergent using the time-averaged amplitude spectra. 2014 Elsevier Inc. All rights reserved.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Discretization of a fractional order ratio-dependent functional response predator-prey model, bifurcation and chaos

This paper deals with a ratio-dependent functional response predator-prey model with a fractional order derivative. The ratio-dependent models are very interesting, since they expose neither the paradox of enrichment nor the biological control paradox. We study the local stability of equilibria of the original system and its discretized counterpart. We show that the discretized system, which is...

متن کامل

Dynamical behavior and synchronization of chaotic chemical reactors model

In this paper, we discuss the dynamical properties of a chemical reactor model including Lyapunov exponents, bifurcation, stability of equilibrium and chaotic attractors as well as necessary conditions for this system to generate chaos. We study the synchronization of chemical reactors model via sliding mode control scheme. The stability of proposed method is proved by Barbalate’s lemma. Numeri...

متن کامل

Study on stability analysis of distributed order fractional differential equations with a new approach

The study of the stability of differential equations without its explicit solution is of particular importance. There are different definitions concerning the stability of the differential equations system, here we will use the definition of the concept of Lyapunov. In this paper, first we investigate stability analysis of distributed order fractional differential equations by using the asympto...

متن کامل

Hybrid Control to Approach Chaos Synchronization of Uncertain DUFFING Oscillator Systems with External Disturbance

This paper proposes a hybrid control scheme for the synchronization of two chaotic Duffing oscillator system, subject to uncertainties and external disturbances. The novelty of this scheme is that the Linear Quadratic Regulation (LQR) control, Sliding Mode (SM) control and Gaussian Radial basis Function Neural Network (GRBFNN) control are combined to chaos synchronization with respect to extern...

متن کامل

Numerical Experiments on the Interaction Between the Large- and Small-Scale Motions of the Navier-Stokes Equations

We consider solutions to the unforced incompressible Navier–Stokes equations in a ¨ ©-periodic box. We split the solution into two parts representing the large-scale and small-scale motions. We define the large-scale as the sum of the first Fourier modes in each direction, and the small-scale as the sum of the remaining modes. We attempt to reconstruct the small-scale by incorporating the large...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:
  • Applied Mathematics and Computation

دوره 248  شماره 

صفحات  -

تاریخ انتشار 2014