A numerical approach to hopf bifurcation points

The computation of Hopf bifurcation points for differential equations with a parameter α is considered. It is well known that the Hopf bifurcation is well characterized by the fact that the first (largest) Lyapunov exponent is zero and the rest are not zero, or by the fact that the corresponding Jacobian has two purely imaginary roots which cross the imaginary axis as α increases through some α₀. An algorithm for detecting Hopf bifurcation points is given in this paper. The basic ideas and techniques are exemplified for the Kuramoto-Sivashinsky equations. These equations are chosen because they are fairely simple while the dynamics is sufficiently complicated. Project supported by the Fundation of Science and Technology of Shanghai Education Commission