In this paper,the relaxation oscillation of a slow-fast Rosenzweig-MacArthur model is investigated.The existence of the relaxation oscillation cycle is obtained by means of geometric singular perturbation theory and e...In this paper,the relaxation oscillation of a slow-fast Rosenzweig-MacArthur model is investigated.The existence of the relaxation oscillation cycle is obtained by means of geometric singular perturbation theory and entry-exit function.The asymptotical stability of the cycle which follows from the negativity of the Floquet multiplier further guarantees the uniqueness of the relaxation oscillation.The theoretical results are verified by numerical simulations.展开更多
This paper presents an investigation on the phenomenon of delayed bifurcation in time-delayed slow-fast differential systems.Here the two delayed's have different meanings.The delayed bifurcation means that the bi...This paper presents an investigation on the phenomenon of delayed bifurcation in time-delayed slow-fast differential systems.Here the two delayed's have different meanings.The delayed bifurcation means that the bifurcation does not happen immediately at the bifurcation point as the bifurcation parameter passes through some bifurcation point,but at some other point which is above the bifurcation point by an obvious distance.In a time-delayed system,the evolution of the system depends not only on the present state but also on past states.In this paper,the time-delayed slow-fast system is firstly simplified to a slow-fast system without time delay by means of the center manifold reduction,and then the so-called entry-exit function is defined to characterize the delayed bifurcation on the basis of Neishtadt's theory.It shows that delayed Hopf bifurcation exists in time-delayed slow-fast systems,and the theoretical prediction on the exit-point is in good agreement with the numerical calculation,as illustrated in the two illustrative examples.展开更多
文摘In this paper,the relaxation oscillation of a slow-fast Rosenzweig-MacArthur model is investigated.The existence of the relaxation oscillation cycle is obtained by means of geometric singular perturbation theory and entry-exit function.The asymptotical stability of the cycle which follows from the negativity of the Floquet multiplier further guarantees the uniqueness of the relaxation oscillation.The theoretical results are verified by numerical simulations.
基金supported by the Foundation for the Author of National Excellent Doctoral Dissertation of China (Grant No.200430)in part by the National Natural Science Foundation of China (Grant Nos.10825207,10532050)
文摘This paper presents an investigation on the phenomenon of delayed bifurcation in time-delayed slow-fast differential systems.Here the two delayed's have different meanings.The delayed bifurcation means that the bifurcation does not happen immediately at the bifurcation point as the bifurcation parameter passes through some bifurcation point,but at some other point which is above the bifurcation point by an obvious distance.In a time-delayed system,the evolution of the system depends not only on the present state but also on past states.In this paper,the time-delayed slow-fast system is firstly simplified to a slow-fast system without time delay by means of the center manifold reduction,and then the so-called entry-exit function is defined to characterize the delayed bifurcation on the basis of Neishtadt's theory.It shows that delayed Hopf bifurcation exists in time-delayed slow-fast systems,and the theoretical prediction on the exit-point is in good agreement with the numerical calculation,as illustrated in the two illustrative examples.
