Hopf bifurcation and strange attractors in Chebyshev spectral solutions of the Burgers equation
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.
Chebyshev pseudospectral collocation method
scientific article; zbMATH DE number 847548
scientific article