Canard Phenomenon in an SIRS Epidemic Model with Nonlinear Incidence Rate

In this paper, we propose an SIRS epidemic model with a new complex nonlinear incidence rate, which describes the psychological effect of some diseases on the community as the number of infective individuals increases, including linear and nonlinear hazards of infection. The canard phenomenon for the model is analyzed, and its epidemiological meaning is discussed. By using geometrical singular perturbation theory and blow up technique, we investigate the relaxation oscillation of the model with the special fold point [Formula: see text]. The unique existence of the limit cycle is proved. We verify the existence of the canard cycle without head by using singular perturbation theory and analyze the cyclicity of the limit cycle. The detailed formula for slow divergence integral of the model is presented. We also discuss and prove the existence of the canard cycle with head. Numerical simulations are done to demonstrate our theoretical results.

Emergence of localized patterns in globally coupled networks of relaxation oscillators with heterogeneous connectivity

Relaxation oscillators may exhibit small amplitude oscillations (SAOs) in addition to the typical large amplitude oscillations (LAOs) as well as abrupt transitions between them (canard phenomenon). Localized cluster patterns in networks of relaxation oscillators consist of one cluster oscillating in the LAO regime or exhibiting mixed-mode oscillations (LAOs interspersed with SAOs), while the other oscillates in the SAO regime. We investigate the mechanisms underlying the generation of localized patterns in globally coupled networks of piecewise-linear (PWL) relaxation oscillators where global feedback acting on the rate of change of the activator (fast variable) involves the inhibitor (slow variable). We also investigate of these patterns are affected by the presence of a diffusive type of coupling whose synchronizing effects compete with the symmetry breaking global feedback effects.

Canard Limit Cycle of the Holling-Tanner Model

By using the singular perturbation theory on canard cycles, we investigate the canard phenomenon for the Holling-Tanner model with the intrinsic growth rate of the predator small enough. The obtained result shows that there may be at most one canard limit cycle, and the range of small parameters is estimated. The phenomenon of outbreak is explained.