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For a positive integer n and R>0, we set $$B_R^n=\left\{x\in {ℝ}^n|{∥x∥}_{\infty }<R\right\}$$ . Given R>1 and n≥4 we construct a sequence of analytic perturbations (H j) of the completely integrable Hamiltonian $$h\left(r\right)=\frac{1}{2}{r}_1^2+...\frac{1}{2}{r}_{n-1}^2+r_n$$ on $${𝕋}^n\times B_R^n$$ , with unstable orbits for which we can estimate the time of drift in the action space. These functions H j are analytic on a fixed complex neighborhood V of $${𝕋}^n\times B_R^n$$ , and setting $${\epsilon }_j:={∥h-H_j∥}_{C^0\left(V\right)}$$ the time of drift of these orbits is smaller than (C(1/ɛ j)1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly resonant surface,...