<p>This paper appears as the confluence of hyperbolic dynamics, symplectic topology and low-dimensional topology. We show that composite symplectic Dehn twists have a certain form of nonuniform hyperbolicity: it has positive topological entropy as well as two families of local stable and unstable Lagrangian manifolds, which are analogous to signatures of pseudo-Anosov mapping classes. Moreover, we show that the rank of the Floer cohomology group of these compositions grows exponentially under iterations, and provide a classification of the symplectic mapping class group of the <i>A</i><Stack> _<i>m</i> ² </Stack> configuration, which partially answers a question of Smith concerning the classification of the symplectic mapping class group in higher-dimensions. Finally, we propose a conjecture on the positive metric entropy of our model and point out its relationship with the standard map.</p>