Predicting multistability of parameterized time-delay dynamical systems using reservoir computing

Parameterized time-delay systems exhibit rich dynamics, with multistability as a typical phenomenon. This leads to multiple bifurcation diagrams, as the system's asymptotic state depends critically on initial conditions, resulting in distinct evolutionary paths under parameter variation. Predicting the multistability is essential for understanding the system's global behavior. Reservoir computing, an efficient machine learning model widely used in dynamics prediction, is employed here to address this challenge. To capture the system's intrinsic characteristics, we train the model with data from various parameter values and initial functions. For two systems, one with dual Hopf bifurcation diagrams and the other with dual period-doubling bifurcation diagrams, the model yields prediction error rates of 0.215% and 0.033%, respectively. The numerical results demonstrate that the complex dynamics exhibited by parameterized time-delay systems can be effectively predicted using the reservoir computing approach. This study thus provides a framework for extending the application of reservoir computing to intricate, multistable dynamical systems.