Two-dimensional invariant manifolds and global bifurcations: Some approximation and visualization studies

We illustrate and discuss the computer-assisted study (approximation and visualization) of two-dimensional (un)stable manifolds of steady states and saddle-type limit cycles for flows in Rⁿ. Our investigation highlights a number of computational issues arising in this task, along with our solutions and "quick-fixes" for some of these problems. Two examples illustrative of both successes and shortcomings of our current approach are presented. Representative "snapshots" demonstrate the dependence of two-dimensional invariant manifolds on a bifurcation parameter as well as their interactions. Such approximation and visualization studies are a necessary component of the computer-assisted study and understanding of global bifurcations.