Data-driven model identification near a supercritical Hopf bifurcation using phase-based approaches

A data-driven model identification strategy is developed for dynamical systems near a supercritical Hopf bifurcation with nonautonomous inputs. This strategy draws on phase–amplitude reduction techniques, analytically relating the phase and amplitude response curves to the terms of the controlled Hopf normal form. Fitting can be performed by recording the system output during the relaxation to the stable limit cycle after applying as few as two carefully timed pulse inputs. Unlike standard phase-based model identification strategies, the resulting model is valid in the neighborhood of the Hopf bifurcation, rather than just in a close vicinity of the unperturbed limit cycle. This strategy is illustrated in two examples with relevance to circadian oscillations. In each example, the proposed model identification strategy allows for the formulation, solution, and implementation of a closed loop nonlinear optimal control problem.