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We are interested in the following differential equation $\ddot u(t) = -Au(t) - c A \dot u(t) + \lambda u(t) + F(u(t))$ where $c > 0$ is a dampingfactor, $A$ is a sectorial operator and $F$ is a continuous map. We considerthe situation where the equation is at resonance at infinity, which means that$\lambda$ is an eigenvalue of $A$ and $F$ is a bounded map. We providegeometrical conditions for the nonlinearity $F$ and determine the Conley indexof the set $K_\infty$, that is the union of the bounded orbits of thisequation.

Homotopy invariants methods in the global dynamics of strongly damped wave equation