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Codimension n saddle-nodes

We consider the generic unfolding of a diffeomorphism on a compact C∞ manifold that is Morse–Smale except for one non-hyperbolic periodic orbit being a codimension n saddle-node (one eigenvalue is 1, the other eigenvalues have norm different from 1). Local and global bifurcations are described. We characterize structural stability of the unfolding, depending on the codimension. A universal model of the unfolding is given when there is stability. Dynamical behaviour is analysed in other cases.