The persistence and the transversality of orbits heteroclinic (or homoclinic) to normally hyperbolic invariant manifolds are studied. The obtained results extend and improve some classical results (Wiggins and Yamashita) in the theory of dynamical systems on smooth manifolds. The author considers the following \(C^r\)-smooth system \[ \dot x=f(x,y)+\varepsilon h(x,y,z,\mu,\varepsilon),\quad \dot y=\varepsilon g(x,y,z,\mu,\varepsilon),\quad \dot z=w(x,y)+\varepsilon v(x,y,z,\mu,\varepsilon), \] where \(x\in\mathbb{R}^n\), \(y\in\mathbb{R}^m\), \(z\in \{(z_1,\dots,z_l):z_i=z_i+T\), \(z_i\in\mathbb{R}\}\). Under some appropriate conditions, the author shows that there exists a heteroclinic orbit of the system to hyperbolic invariant tori. Moreover, some results for transversality of stable and unstable manifolds of the tori are considered. Some applications for weakly coupled oscillators of nonlinear circuits are obtained.