Using a phase-space cross section to study the structure of phase boundaries in large complex systems

For large coupled nonlinear systems, it is difficult to visualize the high-dimensional phase space, which has been thoroughly studied in smaller systems with regard to phenomena such as riddled basins. Here, we propose a method to probe the phase space by defining a phase-space cross section. The method is applied to a system of dynamically coupled maps introduced by Ito and Kaneko ( 2001 Phys. Rev. Lett. 88 028701, 2003 Phys. Rev. E 67 046226). We show that the transitions between phases of different synchronization behaviour are not always sharp, but can be characterized by fractal boundaries in both phase and parameter space.