Discrete damped or undamped gradient systems described by nonlinear au
tonomous ordinary differential equations (ODE) of motion are examined
in detail. Emphasis is given to the relationship between static with p
ossibly existing dynamic bifurcations. Criteria for dynamic bifurcatio
ns and stability of precritical, critical, and postcritical states ass
ociated with the nature of Jacobian eigenvalues are also presented. Ca
ses of discrepancies between local and global dynamic analysis regardi
ng the stability of equilibriums are reported for the first time. Fina
lly, using a simple energy criterion, exact dynamic buckling loads for
vanishing but nonzero damping are readily obtained.