Maximal Lyapunov exponent at Crises

We study the variation of Lyapunov exponents of simple dynamical systems near attractor-widening and attractor-merging crises. The largest Lyapunov exponent has universal behaviour, showing abrupt variation as a function of the control parameter as the system passes through the crisis point, either in the value itself, in the case of the attractor-widening crisis, or in the slope, for attractor merging crises. The distribution of local Lyapunov exponents is very different for the two cases: the fluctuations remain constant through a merging crisis, but there is a dramatic increase in the fluctuations at a widening crisis.