Bifurcations For Periodically Perturbed Systems

Bifurcations of periodically perturbed systems are studied in this thesis.We discuss the effect of periodic perturbation on systems via different periodic mechanisms.For a periodically perturbed predator-prey system with a nonmonotonic generalized Holling type IV functional response,we consider four periodic mechanisms by a continuation technique.That is,the system is perturbed with periodically varying natural death rate,periodically varying carrying capacity,periodically and non-synchronously varying death rate and carrying capacity with the same period,and periodically and non-synchronously varying death rate and carrying capacity with different periods.Firstly,when the natural death rate is periodically varied,we get six different bifurcation diagrams corresponding to different bifurcation cases of the unforced system.If the carrying capacity is periodic,two different bifurcation diagrams are obtained.Here we cannot get a “universal diagram” like the one in the periodically perturbed system with monotonic Holling type II functional response;that is,the two elementary periodic mechanisms have different effects on the population.Secondly,when both the natural death rate and the carrying capacity are perturbed with two different periodic mechanisms,the phenomena that arise are to some extent different.The bifurcation results also show that each periodic mechanism can display complex dynamics such as multiple attractors including stable cycles of different periods,quasi-periodic solutions,chaos,switching between these attractors and catastrophic transitions.In addition,we give some orbits in phase space and corresponding Poincaré sections to illustrate different attractors.Then,a generalization of a Burridge-Knopoff spring-block model is investigated to illustrate the earthquake fault system.The model can undergo Hopf bifurcation and Fold bifurcation of limit cycles.Considering the cyclical nature of the spring strength,the model with periodic perturbation is further explored via a continuation technique and numerical bifurcation analysis.It is shown that the periodic perturbation induces abundant dynamics – the existence,the switch,and the coexistence of multiple attractors including periodic solutions with various periods,quasi-periodic solutions,chaotic solutions through torus destruction or cascade of period doublings.Throughout the results obtained,one can see that the Burridge-Knopoff spring-block model will be very complex when it is periodically perturbed,and even very small variation of a parameter can result in radical changes of the dynamics,which provides a new insight into the earthquake fault system.Moreover,the effect of periodic perturbation is considered on a system which undergoes a cusp bifurcation.It is found that,for small magnitude of perturbation,the steady-state solutions of the unperturbed system are transformed into periodic solutions.The periodically perturbed systems can have new fold and cusp bifurcations of periodic solutions,corresponding to bifurcations of the steady-state solutions in the unperturbed systems;and the deviation degree of the new bifurcations from the original ones is closely related to the magnitude of perturbation and the perturbation functions.By the numerical bifurcation analysis,several example systems are discussed to illustrate the theoretical results.