MK's abstract--
Recent interest in epidemical model is dynamical behaviours of SIR with seasonal forcing. The general properties of the model with seasonal forcing are:
i) when the forcing is small, the dynamics of infected hosts tend to be a year cycle which is the same period with seasonal forcing.
ii) when we have stronger forcing, the dynamics becomes period of two years, four years, eight years and the dynamics becomes chaotic eventually, i.e. there is a period doubling bifurcation.
iii) new periods also appear in order of the smaller integers by saddle node bifurcation (sometimes called "period adding").
iv) especially when the seasonal forcing is large, the periods which appear by the saddle node bifurcation break the chaotic dynamics and there is a "window between intervals of chaos".
v) to understand the relationships between the different stable attractors we use the method of Poincare maps to reveal otherwise hidden unstable attractors.

JG's abstract---
Abstract.
Analysing dynamical models is complicated by the multiplicity of states that can be excited in the interaction between model nonlinearity and external forcing. This multiplicity, typically created by a hierarchy of subcritical subharmonics, can lead to high amplification of the forcing signal and complex switching under chaos and stochastic externalities. Our objective is to understand the structure of these possible modes of system oscillation, in particular the conditions under which they appear and disappear. The analysis is carried out by transforming the problem into a problem of locating the resonances of the system under variable period forcing. The hierarchy of resonances identified has a well defined generic structure based on the natural period of oscillation of the model, whether stable or unstable when isolated.
The methodology will be applied to a study of (i) childhood diseases and will provide an explanation for why different diseases behave in different ways (ii) pathogen virulence with RHD in rabbits as an example (iii) reinforcement and interference in predator-prey models (iv) the effect of age structure in delay-differential equations with "Plodia" as an example (v) the interaction of two pathogen strains (Kamo and Sasaki, 2002).