The evolution of the invariants (RandQ) of the velocity gradient tensor in homogeneous isotropic turbulence is investigated using data from direct numerical simulation (DNS). The concepts of conditional average time rate of change of the invariants and conditional mean trajectories (CMT) in invariant phase space are introduced to study the dynamics of this flow. The resulting dynamical system in the(R,Q)phase space is a clockwise spiral with a stable focus at the origin, illustrating that in the mean, the cyclic sequence of topological evolution following a fluid particle is unstable-node/saddle/saddle (UN/S/S)→stable-node/saddle/saddle (SN/S/S)→stable-focus/stretching (SF/S)→unstable-focus/contracting (UF/C). The mean rates of change ofRandQ, i.e.,R˙,Q˙,are found to be negligible near the right branch of the null discriminant(D=0)curve, indicating that this curve is an attractor in the(R,Q)space. The effects of both the diffusion term and the anisotropic part of the pressure Hessian term on the dynamics of the invariants have also been analyzed using the conditional averages. Both contributions are found to be important in the dynamics of the velocity gradient invariants. Based on these results the extent of the validity of the model equations governing the evolution ofRandQproposed by Cantwell [Phys. Fluids A4, 782 (1992)] and Dopazo &etal; [“Velocity gradients in turbulent flows. Stochastic models,” Ninth Symposium on “Turbulent Shear Flows,” Kyoto, Japan, 1993, pp. 26-2-1–26-2-5] are discussed. ©1998 American Institute of Physics.