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Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcations

The biological models - particularly the ecological ones - must 
be understood through the bifurcations they undergo as the parameters vary. However, the transition between two dynamical behaviours of a same system for diverse values of parameters may be sometimes quite involved. For instance, the analysis of the non generic motions near the transition states is the first 
step to understand fully the bifurcations occurring in complex dynamics. 
  In this article, we address the question to describe and explain a double bursting behaviour occuring for a tritrophic slow–fast system. We focus therefore on the appearance of a double homoclinic bifurcation of the fast subsystem as the predator death rate parameter evolves. 
 
  The first part of this article introduces the slow–fast system which extends Lotka–Volterra dynamics by adding a superpredator. The second part displays the analysis of singular points and bifurcations undergone by fast dynamics. 
The third part is devoted to the flow analysis near the homoclinic points. Finally, the fourth part is concerned with the main results about the existence of periodic orbits of different periods as the two homoclinic orbits are close 
enough to each other.