A cartographer charting a route through a mountain range notes not only peaks and valleys but also passes where the trail sharply changes direction. In dynamical systems, bifurcation analysis plays the role of such a cartographer: it shows where, under a smooth parameter change, the system's behavior qualitatively rearranges itself, for instance, a stable equilibrium gives way to oscillations.

For delay systems, where the rate of change depends on past states, drawing such a map is especially challenging. The delay makes the phase space infinite-dimensional, and analytical methods quickly hit a wall. This is where numerical continuation comes to the rescue: the computer carefully steps along the solution, varying the parameter, and watches for a bifurcation, a moment when a solution is born, disappears, or loses stability.

In the El Niño model with a constant delay, numerical continuation revealed a family of periodic orbits with periods around 3–4 years, consistent with observations of ocean surface temperature. As coupling parameters were varied, bifurcations were found where regular cycles lose stability and transition to more complex behavior. This is as if the cartographer discovered that beyond the pass, the trail does not simply descend but branches into many paths.

In the second example, a scalar equation with two state-dependent delays, the delays are determined by the solution itself. Such a system is even harder to analyze, but DDE-BIFTOOL managed: several stable equilibria and periodic solutions were found, as well as bifurcations where the system can abruptly switch between regimes. Key bifurcations occurred at parameter values around 0.5–1.0.

Returning to the cartographer metaphor: numerical continuation does not merely draw lines on a map; it explains why the route goes the way it does. For climate models, neuroscience, and engineering, where delay plays a key role, such a tool allows predicting when a system might abruptly change regime and preparing for the pass in advance.