Hybrid methods in the study of the oscillatory dynamics in complex systems

Formulae for obtaining bifurcation curves of dynamical systems described by a set of ordinary differential equations (ODE) are introduced. The plot of bifurcation curves in the plane of characteristic parameters gives certain areas having distinguished local bifurcation diagrams obtainable by standard tools of bifurcation theory. A trade-off between the ease of obtaining local (but incomplete) results and the complexity of finding global diagrams is analyzed, and a modified scheme to capture different configurations of local bifurcation diagrams and, at the same time, obtain global results for each case of interest is proposed. This makes it possible to explore, in a great detail, the bifurcation structure of larger regions in the parameter plane. Briefly, the procedure consists in detecting the bifurcation curves and, in the case of Hopf bifurcations, in calculating the stability of the emerging limit cycles. After that, a detailed study by means of AUTO package is performed. The first method yields a gross distinction among neighboring areas with different local bifurcation diagrams, while the second method yields the global bifurcation structure in a one-parameter plot. The combined hybrid procedure is a fast computational tool for analyzing the dynamic behaviour in general systems.