Canard explosion in chemical and optical systems

The paper deals with the study of the relation between the Andronov--Hopf bifurcation, the canard explosion and the critical phenomena for the van der Pol's type system of singularly perturbed differential equations. Sufficient conditions for the limit cycle birth bifurcation in the case of the singularly perturbed systems are investigated. We use the method of integral manifolds and canards techniques to obtain the conditions under which the system possesses the canard cycle. Through the application to some chemical and optical models it is shown that the canard point should be considered as the critical value of the control parameter.