Homoclinic and stable periodic solutions for differential delay equations from physiology

Mackey-Glass type of differential delay equations with one-parameter family of hump-shaped nonlinear functions is considered. The existence of a homoclinic solution for suitable nonlinear function is proved. As far as we know the possibility of such a homoclinic solution for this type of equations had never been discussed before. Also the existence of stable periodic solutions for values of parameter close to the homoclinic one is studied. An example of nonlinear functions satisfying all suffcient conditions of our theoretical results is given. Numerical approximations of solutions for different values of parameter are shown.