Dynamic analysis of a double Hopf 1:2 resonance in an delay differential equation via a frequency-domain method

GRISELDA R. ITOVICH; FRANCO S. GENTILE; JORGE L. MOIOLA

Río de Janeiro

Congreso; International Congress of Mathematicians (ICM 2018); 2018

It is considered a second order differential equation with one delay anda quadratic nonlinearity, which includes three additional parameters. Thismodel exhibits two equilibrium points, whose stability was analyzed com-pletely. Besides, some particular parameter confi…gurations were found wheresome different resonant double Hopf bifurcations take place, in particularof type 1:2. It is known that in a neighborhood of this singularity, cycleswith frequency ! or 2! appear singly but also simultaneously. Moreover,the existence of period doubling bifurcations of cycles is frequent in the de-scribed context. Related with dynamic systems control, the frequency do-main methodology allows, via the graphical Hopf theorem, the detection ofHopf bifurcations and the attainment of approximate expressions for the ris-ing periodic solutions, through harmonic balance. Thus, different dynamicapproaches were analyzed in the unfolding of this singularity like: the num-ber of existing limit cycles associated to one or other frequency as well asthe cycles stability over the Hopf bifurcations curves. Also, type saddle-node, period-doubling or torus (or Neimark Sacker) bifurcations of cycleswere detected and its associated curves were obtained in some parameterplane. These results were established starting from fourth order approxi-mations of the periodic solutions, coming through the selected methodology,one Tchebyschev collocation method to build a …finite approximation of themonodromy operator and the later calculus of relevants Floquet multipliers.All the achieved results were checked with those coming from well-knownsoftwares for delay differential equations, showing the local effectiveness ofthe used method.