On Trigonometric Collocation in Hopf Bifurcation
Eckart W. Gekeler · Birkhäuser Basel eBooks · 1992
Trigonometric collocation methods are used to approximate simple Hopf bifurcation problems, wy’ + f(y,μ) = 0, with 2π-periodic solutions. The discrete system is solved by a constructive Ljapunov-Schmidt reduction originally due to Keller and Langford. In this method no component of the solution is fixed in advance therefore codes for the Fast Fourier Transform can be applied in the iteration which is of the form $$ F_z (z_0, 0)(z_{new} - z) = F(z,\varepsilon ),{\text{ }}|\varepsilon |small $$ This way yields a fast algorithm with low storage requirements for the matrix Fz(z0,0).