Hopf bifurcation and strange attractors in Chebyshev spectral solutions of the Burgers equation
type: Item
instance of: scholarly article
title: Hopf bifurcation and strange attractors in Chebyshev spectral solutions of the Burgers equation
zbMATH Open document ID: 0844.65074
DOI: 10.1016/0096-3003(94)00242-8
author: H. Dang-Vu
author: C. Delcarte
published in: Applied Mathematics and Computation
publication date: 1996-08-29T00:00:00Z
review text: Two spectral methods, namely the ``Chebyshev pseudospectral collocation method'' and the ``Chebyshev tau spectral method'' are considered for the Burgers equation \[ u_t= - uu_x+ \nu u_{xx}+ f(x),\quad -1\leq x\leq 1,\quad t\geq 0,\quad \nu> 0, \] with boundary and initial conditions \(u(- 1, t)= u(1, t)= 0\), \(u(x, 0)= u(x)\). The resulting system of ordinary differential equations is numerically solved by aid of the Adams-Moulton method. The function \(u_0(x, t)= \sin(ux)\) is a steady state solution if \(f(x)= \pi(\cos(\pi x)+ \pi\tau)\sin (\pi x)\). By systematically carrying out numerical case studies the authors investigate questions of stability of \(u_0\), of Hopf bifurcation, limit cycles and strange attractors.
Mathematics Subject Classification ID: 65M70
Mathematics Subject Classification ID: 65M20
Mathematics Subject Classification ID: 35Q53
zbMATH DE Number: 847548
zbMATH Keywords: Chebyshev pseudospectral collocation method
zbMATH Keywords: Chebyshev tau spectral method
zbMATH Keywords: Burgers equation
zbMATH Keywords: Adams-Moulton method
zbMATH Keywords: stability
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: limit cycles
zbMATH Keywords: strange attractors
reviewed by: Rudolf Gorenflo
MaRDI profile type: Publication
full work available at URL: 0096-3003(94)00242-8
OpenAlex ID: W2062483422
cites work: Q3762690
cites work: Note concerning our paper ''On the nature of turbulence''. Commun. math. Phys. 20, 167-192 (1971)
cites work: Q1233671
cites work: Q3930015
cites work: An accurate solution of the Poisson equation by the Chebyshev collocation metod
cites work: Q4880406
cites work: Determining Lyapunov exponents from a time series
cites work: A five-dimensional truncation of the plane incompressible Navier-Stokes equations
Recommended article: Q4892079
Recommended article: Dynamics in spectral solutions of Burgers equation
Recommended article: Q3782702
Recommended article: Strange attractor of KdV-Burgers equation
Recommended article: Q4025735
Recommended article: Dynamics on Spectral Solutions of Forced Burgers Equation
Recommended article: Q4880406
Recommended article: Q4870858
Recommended article: Q5398539
Recommended article: Q4937623
description: scientific article; zbMATH DE number 847548