  We study the problem of instability in the following a priori unstable
Hamiltonian system with a time-periodic perturbation
\[\mathcal{H}_\varepsilon(p,q,I,\varphi,t)=h(I)+\sum_{i=1}^n\pm
\left(\frac{1}{2}p_i^2+V_i(q_i)\right)+\varepsilon H_1(p,q,I,\varphi, t), \]
where $(p,q)\in \mathbb{R}^n\times\mathbb{T}^n$,
$(I,\varphi)\in\mathbb{R}^d\times\mathbb{T}^d$ with $n, d\geq 1$, $V_i$ are
Morse potentials, and $\varepsilon$ is a small non-zero parameter. Using
geometric methods we prove that Arnold diffusion occurs for generic analytic
perturbations $H_1$. Indeed, the set of admissible $H_1$ is $C^\omega$ dense
and $C^3$ open. The proof also works for arbitrarily small $V_i$. Our
perturbative technique for the genericity is valid in the $C^k$ topology for
all $k\in [3,\infty)\cup\{\infty, \omega\}$.
