Summary: A perturbation method based on multiple scales is used for computing the normal forms of nonlinear dynamical systems. The approach, without the application of center manifold theory, can be used to systematically find the explicit normal form of a system described by a general \(n\)-dimensional differential equation. The attention is focused on the dynamic behaviour of a system near a critical point characterized by two pairs of purely imaginary eigenvalues without resonance. The method can be easily formulated and implemented using a computer algebra system. Maple programs have been developed which can be ‘automatically’ executed by a user without knowing computer algebra. Examples chosen from mathematics, electrical circuits, mechanics and chemistry are presented to show the applicability of the technique and the convenience of using computer algebra.