Delays in differential equations reflect the finite speed of signal transmission in real systems, from climate feedbacks to neural networks. However, analytical study of such equations is hampered by the infinite-dimensional nature of the phase space. This work sits somewhat apart: in the archive, it is surrounded by studies on neural-network cartographers and hidden order in leaderless reorganizations, but direct neighbors on delay dynamical systems are scarce. Nevertheless, the overarching theme, uncovering structure in complex nonlinear processes, unites these works. This chapter fills a gap by showing how numerical bifurcation analysis implemented in DDE-BIFTOOL allows systematic investigation of DDEs, including the case of state-dependent delays.
The main tool is the DDE-BIFTOOL package, designed for numerical continuation of solutions and bifurcations of delay differential equations. The package can continue steady states (fixed points) and periodic orbits with respect to one or two parameters, detect codimension-one bifurcations (such as saddle-node or Hopf bifurcations) and some codimension-two bifurcations, and compute their normal forms. To demonstrate its capabilities, two model problems are considered: a conceptual DDE model of the El Niño phenomenon with constant delay, and a scalar DDE with two state-dependent feedbacks. In the latter case, the delays are determined by the solution itself, which complicates the analysis and requires special handling in the package.
For the El Niño model, numerical continuation revealed a family of periodic orbits corresponding to oscillations in ocean surface temperature and allowed tracking their stability as coupling parameters change. Bifurcations were found where the periodic solution loses stability, corresponding to a transition from regular El Niño cycles to more complex behavior. In the scalar DDE with two state-dependent delays, the analysis showed the existence of multiple stable equilibria and periodic solutions, as well as bifurcations where the system can switch abruptly between regimes. Key numbers: in the El Niño model, the oscillation period is about 3–4 years, consistent with observations; in the scalar example, bifurcations occur at parameter values around 0.5–1.0.
This work demonstrates that modern numerical bifurcation analysis methods are applicable to a wide class of delay systems, including those with state-dependent delays. This distinguishes it from neighboring works in the archive that focus on law recovery by neural networks or analysis of leaderless reorganizations: here the emphasis is on direct computation of bifurcation diagrams and normal forms. Such an approach allows not only finding solutions but also understanding how they are born, disappear, and lose stability as parameters vary, which is critical for predicting the behavior of real systems.
Further development of DDE-BIFTOOL is expected to include support for more complex types of delays (for example, distributed delays), automatic detection of higher-codimension bifurcations, and integration with machine learning methods to speed up computations. Application of the package to problems in neurobiology, climatology, and engineering, where delay plays a key role, is also promising.
The results will be useful for researchers working with models of climate, neural networks, population dynamics, and control systems, where delay significantly affects dynamics.
Immediate next steps include expanding the library of examples, documenting new package features, and conducting comparative studies with other DDE tools.
In the field of dynamical systems, the complete classification of bifurcations in systems with state-dependent delays remains an open question, as does the connection between local bifurcations and global chaotic behavior. This work provides a tool for systematic investigation of such problems.
DDE-BIFTOOL has been under development since the 1990s and is still actively used by dozens of research groups worldwide; its core is written in MATLAB and is compatible with GNU Octave.