The author considers the following time-periodic perturbation of the \(n\)-dimensional autonomous system \[ \dot{x} = F(x) + G(t,x,\alpha) \] with \(x\in{\mathbb{R}}^{n}\), \(\alpha\in{\mathbb{R}}^{k}\), \(t\in{\mathbb{R}}\) and where \(F:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\) and \(G: \mathbb{R}\times \mathbb{R}^{n}\times \mathbb{R}^{k} \rightarrow \mathbb{R}^{n}\) are sufficiently smooth, \(G(t,x,0) = 0\) and \(G(t+\omega,x,\alpha) = G(t,x,\alpha)\) \(\forall t\in{\mathbb{R}}\).
It is also supposed that the unperturbed system \(\dot{x} = F(x)\) has a nonhyperbolic and noncritical closed orbit \(\Gamma_{0} = \{u_{0}(t): 0 \leq t<T \}\) in the phase space which is interpreted as a nonhyperbolic invariant torus \(\mathbb{T}_{0}^{2} = \Gamma_{0} \times \mathbb{S}^{1}\).
Let us denote by \(s\) the order of the Jordan block associated to the eigenvector \(\dot{u}_{0}(0)\) of the unit characteristic multiplier of \(\Gamma_{0}\) in the monodromy matrix \(X(t)\) for the linear variational equation \[ \dot{x} = DF(u_{0}(t))x. \] When \(s=1\) the nonhyperbolic closed orbit is noncritical and critical if \(s>1\).
Under some technical conditions which depend only on the original system, it is proved that classical bifurcations (such as saddle-node, transcritical and pitchfork) to a nonhyperbolic but noncritical invariant torus, appear.
Two interesting examples are stated. One is a system of differential equations presenting a saddle-node bifurcation of a nonhyperbolic, invariant torus, and the other presenting a transcritical bifurcation.
Since testing that a closed orbit is nonhyperbolic and noncritical is difficult, an interesting sufficient condition on the unperturbed system is given to get it. Such condition is: let \(\Gamma_{0}\) a closed orbit of period \(T\) of the unperturbed system, then if the linear variational equation has exactly two linear independent T-periodic solutions, \(\Gamma_{0}\) is nonhyperbolic and noncritical.