Abstract: The authors study bifurcations from a heteroclinic manifold connecting two non-hyperbolic equilibrium P₀ and P₁ for a n-dimensional dynamical system. They show that under some assumptions, each equilibrium P_i splits into two equilibria P_i and P_i(α), i=0, 1, and find the Melnikov vector conditions assuring the existence of a heteroclinic orbit from P₁ (α) to P₀ (α) along directions that are tangent to the strong unstable (resp.strong stable) manifold of P₁ (α) (resp.P₀(α)). The exponential trichotomy and the unified and geometrical method are used to prove their results.

Key words: nonhyperbolic equilibrium, heteroclinic manifold, exponential trichotomy