The authors consider a family of local diffeomorphisms \(G(x,\mu)\), \(G(x_ 0,\mu_ 0)=x_ 0\), where i) \(G: U\to\mathbb{R}^ n\), \(U\) is a neighbourhood of \((x_ 0,\mu_ 0)\) in \(\mathbb{R}^ n\times\mathbb{R}^ k\); ii) \(D_ x(x_ 0,\mu_ 0)\) has a single eigenvalue of \(-1\) and no other eigenvalues on the unit circle; iii) on its one-dimensional center manifold corresponding to \(-1\), the map \(G(x,\mu_ 0)\) is \(C^{2k+1}\)-conjugate to a \(C^{2k+1}\)-map of the form \(y\to -y+cy^{2k+1}+o(y^{2k+1})\), \(c\neq 0\). A problem of the conjugacy of such a family to the \(k\)-parameter unfolding \(x_ 1\to f_ \varepsilon(x_ 1)=-(\varepsilon_ 1+1)x_ 1-\varepsilon_ 2x^ 3_ 1-\dots-\varepsilon_ k x_ 1^{2k- 1}+cx_ 1^{2k+1}\), is considered. Let \(g(x_ 1,\mu)\) be the center manifold representation of \(G(x,\mu)\). The main result of the paper states that generically there exists a diffeomorphism \(\Psi(x_ 1,\mu)=(Z(x_ 1,\mu),\psi(\mu))\) such that for any fixed \(\mu\), \(g_ \mu(x_ 1)=g(x_ 1,\mu)\) and \(f_{\psi(\mu)}(z)\) are topologically conjugate to each other. Also, \(\Psi\) maps fixed points, period-2 points and bifurcation manifolds of \(g\) to fixed points, period-2 points and corresponding bifurcation manifolds of \(f\), respectively. Low codimension bifurcation diagrams and some applications are given.