In this work we present a set-oriented path following method for the computation of relative global attractors of parameter-dependent dynamical systems. We start with an initial approximation of the relative global attractor for a fixed parameter λ0 computed by a set-oriented subdivision method. By using previously obtained approximations of the parameter-dependent relative global attractor we can track it with respect to a one-dimensional parameter λ > λ0 without restarting the whole subdivision procedure. We illustrate the feasibility of the set-oriented path following method by exploring the dynamics in low-dimensional models for shear flows during the transition to turbulence and of large-scale atmospheric regime changes .

In this work we extend the novel framework developed by Dellnitz, Hessel-von Molo, and Ziessler to the computation of finite dimensional unstable manifolds of infinite dimensional dynamical systems. To this end, we adapt a set-oriented continuation technique developed by Dellnitz and Hohmann for the computation of such objects of finite dimensional systems with the results obtained in the work of Dellnitz, Hessel-von Molo, and Ziessler. We show how to implement this approach for the analysis of partial differential equations and illustrate its feasibility by computing unstable manifolds of the one-dimensional Kuramoto--Sivashinsky equation as well as for the Mackey--Glass delay differential equation.