The author studies a bifurcation of homoclinic and heteroclinic orbits in a k-parameter family of \((m+n)\)-dimensional ODES: \(\dot x=f(x)+g(x,\mu)\), \(x\in {\mathbb{R}}^{m+n}\), \(\mu \in {\mathbb{R}}^ k\) (k\(\geq 2)\), where f and g are smooth and \(g(x,0)=0\). Suppose that the system has three saddle equilibria \(0_ i(\mu)\), \(i=1,2,3\), and the unperturbed system \(\dot x=f(x)\) has a heteroclinic orbit connecting \(0_ 1(0)\) and \(0_ 2(0)\) (i.e. an \((0_ 1,0_ 2)\)-connection) and an \((0_ 2,0_ 3)\)- connection simultaneously. Under some assumptions on the eigenvalues of the linearized equation at equilibrium points and on a non-degeneracy condition for the system the author shows that heteroclinic orbits of new type appear besides the persistent ones of the unperturbed system. A bifurcation diagram is given for such families. Some homoclinic bifurcations are also treated including that one producing a twice- rounding homoclinic orbit. This paper is organized as follows: In {\S} 1, the author gives a precise statement of the problem and the main results. He explains the notion of exponential dichotomy in {\S} 2, and using it, he first studies the persistency condition of heteroclinic orbits in {\S} 3. In {\S} 4 he briefly states a lemma, which gives a property of trajectories near a saddle equilibrium. The main theorems are proved in {\S} 5-6 for the cases of non-critical eigenvalues and of critical eigenvalues. The bifurcation diagrams are treated in {\S} 7 as well as a simple example of a two-dimensional system. {\S} 8 is devoted to a study of the bifurcation of the doubling of a homoclinic orbit. Finally, the author makes some concluding remarks in {\S} 9, especially on several related works.