Two new approaches to computing Hopf bifurcation problems (Q1343087)

The authors consider the numerical computation of Hopf bifurcation for ordinary differential equations. Two extended systems are given for the calculation of Hopf bifurcation problems: the first one is very similar to the usually employed extended systems, but replaces the Jakobian by a difference quotient. It is shown that this extended system is regular and has a continuous family of isolated periodic solutions close to the bifurcation point. The second extended system utilizes the phase shift symmetry by half a period of the linearized system at the bifurcation point. By splitting the solution into its even and odd components it is possible to reduce the interval of integration to half the period at the cost of almost doubling the size of the system. The authors claim that the overall effort is reduced by this method. Finally numerical solutions of several test examples are given. It would be interesting to compare the proposed extended systems with the usual ones; at least for differential equations with reflectional symmetries the second approach looks attractive. Since both systems use difference quotients it would be important to know how to avoid large cancellation errors.