By comparison of different parameters associated with a chaotic evolution, we discuss the relation between several populations of inner Solar System bodies. Tancredi (1998) found that the observed sample of Jupiter family comets (JFCs) and near-Earth asteroids (NEAs) have Lyapunov times (the inverse of Lyapunov characteristic exponents) grouped in the range between 50 and 150 yr, however the dynamical evolution is strikingly different. By numerical integrations of the gravitational equations we compute a finite estimate of the Lyapunov characteristic exponent, the so-called Lyapunov characteristic indicators (LCIs). The LCI is found by adding short time contributions, the so-called `local Lyapunov characteristic indicators' (cite{froeschle}). The distribution of the local Lyapunov characteristic indicators (DLI) is invariant within a chaotic region and gives us more complete information about the chaotic behaviour. To compare the DLIs of different objects we compute the first four moments of the distributions. Though JFCs and NEAs have similar LCIs, the analysis of the four moments makes possible the distinction between the two populations, since they occupy separate regions in the moment phase-space. We discuss the origin of the Jupiter family as the result of the splitting of a giant comet several thousand years ago. We simulate that event and integrate the dynamical evolution of the fragments. By comparing the DLIs of the fragments with the present JF objects, we observe a much more compact distribution of moments in the simulations than in the real population. The splitting hypothesis is not a plausible explanation for the origin of JFCs. Neither is there evidence for a few-body splitting event generating ~ 10 comets. Nevertheless, the comparison of DLIs proves to be a useful tool for discussing some splitting events already suggested among observed JFCs.