In this thesis we provide an analytical proof of the existence of a stable periodic orbit contained in the region of coexistence of the three species of a trithrophic chain . The method used consists in analyzing a triple Hopf bifurcation . For some values of the parameters three limit cycles born via this bifurcation . One is contained in the plane when the top-predator is absent . Another one is not contained in the domain of interest where all variables are positive . The third one is contained where the three species coexist . The techniques for proving this results have been introduced in onother article and are based on the averaging theory of second-order . Existence of this triple Hopf bifurcation has been previously discovered numerically by kooij et al . In the averaging theorem we consider systems of the form Suppose that we have a one-parameter family of systems similar to: with the associated family of averaged systems and suppose that top equation undergoes a bifurcation as µ varies. This Theorem implies that if at µ=µ• top equation undergoes a s addle-node or a Hopf bifurcation, then, for µ near µ• and ? sufficiently small, the Poincare map of first equation also undergoes a Saddle-node or a Hopf bifurcation, then by using this Theorem we prove that a triple Hopf bifurcation born from orginal system. In the secound section of thesis we consider the dynamics near an orbit homoclinic to a fixed point of saddle-focus type in a third-order ordinary differential equation. This has been known as the Silnikov phenomenon, since it was first studied by Silnikov 1996. Belyakov point i.e. codim -2 homoclinic bifurcation point where the transition from Saddle-focus point to Saddle of the equilibrium occurs and Belyakov bifurcation is a special codim -2 homoclinic bifurcation also was introduced and we have shown that model undergoes this type bifurcation but Belyakov point can not be considered as origion of chaos in food chain. Finally we have proved below Theorem: Theorem:Consider a generic smooth three-dimensional system of ordinary differential equations depending upon two parameters , having at some parameter values a homoclinic orbit, 1 to an equilibrium O with eigenvalues ?1,2=?• 0 , ?3= ? - ?• . Then the corresponding point in the parameter plane is the origion of three countable sets of subsidiary bifurcation curves, namly: •tn1:tangent bifurcation curves of periodic orbits making one global passage near 1; •fn1:flip bifurcation curves of periodic orbits making one global passage near 1; •h n2:homoclinic bifurcation curves , corresponding to the existence of Saddle-focus homoclinic orbits making two global passage near 1.