Double Hopf bifurcation analysis can be used to reveal some complicated dynamical behavior in a dynamical system, such as the existence or coexistence of periodic orbits, quasi-periodic orbits, or even chaos. In this paper, an algorithm for deriving the normal form near a codimension-two double Hopf bifurcation of a reaction-diffusion system with time delay and Neumann boundary condition is rigorously established, by employing the center manifold reduction technique and the normal form method. We find that the dynamical behavior near bifurcation points are proved to be governed by twelve distinct unfolding systems. Two examples are performed to illustrate our results: for a stage-structured epidemic model, we find that double Hopf bifurcation appears when varying the diffusion rate and time delay, and two stable spatially inhomogeneous periodic oscillations are proved to coexist near the bifurcation point; in a diffusive predator-prey system, we theoretically proved that quasi-periodic orbits exist on two-or three-torus near a double Hopf bifurcation point, which will break down after slight perturbation, leaving the system a strange attractor.

No abstract

No abstract

In this paper, a disease transmission model of SEIR type with stage structure is proposed and studied. Two kinds of time delays are considered: the first one is the mature delay which divides the population into two stages; the second one is the time lag between birth and being able to move freely, which we call the freely-moving delay. Our mathematical analysis establishes that the global dynamics are determined by the basic reproduction number [Formula: see text]. If [Formula: see text], then the disease free equilibrium [Formula: see text] is globally asymptotically stable, and the disease will die out. If [Formula: see text], then a unique positive equilibrium [Formula: see text] exists, and [Formula: see text] is locally asymptotically stable when the freely-moving delay is less than the critical value. We show that increasing this delay can destabilize [Formula: see text] and lead to Hopf bifurcations and stable periodic solutions. By using the normal form theory and the center manifold theory, we derive the formulae for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. Finally, some numerical simulations are carried out to verify the theoretical analysis and some biological implications are discussed.

Heterogeneous delays with positive lower bound (gap) are taken into consideration in Kuramoto oscillators. We first establish a perturbation technique, by which universal normal forms and detailed dynamical behavior of this model can be obtained easily. Theoretically, a hysteresis loop is found near the subcritically bifurcated coherent state on the Ott-Antonsen's manifold. For Gamma distributed delay with fixed variance and mean, we find large gap destroys the loop and significantly increases in the number of coexisted coherent attractors. This result is also explained in the viewpoint of excess kurtosis.