×

Hopf bifurcation and strange attractors in Chebyshev spectral solutions of the Burgers equation. (English) Zbl 0844.65074

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.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Sachdev, P. L., Nonlinear Diffusive Waves (1987), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0624.35002
[2] Ruelle, D.; Takens, F., Commun. Math. Phys., 20, 167-192 (1971) · Zbl 0227.76084
[3] Marsden, J.; McCracken, M., The Hopf Bifurcation and Its Applications (1976), Springer: Springer New York · Zbl 0346.58007
[4] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H., Theory and Applications of Hopf Bifurcation (1981), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0474.34002
[5] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T., Spectral Methods in Fluid Dynamics (1977), Springer: Springer Berlin
[6] Peyret, R., Introduction to Spectral Methods, von Karman Institute Lecture Series 1986-04 (1986), Rhode-Saint Genese, Belgium
[7] Dang-Vu, H.; Delcarte, C., J. Comp. Phys., 104, 211-220 (1993)
[8] Dang-Vu, H.; Delcarte, C., Chebyshev spectral solution of the Burgers equation with high wavenumbers, (Bainov, D.; Covachev, V., Second International Colloquium on Numerical Analysis. Second International Colloquium on Numerical Analysis, Plovdiv University-Bulgaria (1993), VSP International Science Publishers: VSP International Science Publishers Zeist, The Netherlands), 63-72 · Zbl 0845.65056
[9] Wolf, A.; Swift, B.; Swinney, J.; Vastano, J., Physica D, 16, 285-317 (1985)
[10] Boldrighini, C.; Franceschini, V., Commun. Math. Phys., 64, 159-170 (1979)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.