Liapunov exponents and mode-locked solutions for integrate-and-fire dynamical systems

We discuss the notion of Liapunov exponent for integrate-and-fire (IF) type dynamical systems. In contrast to smooth flows there is a contribution to the IF Liapunov exponent arising from the discontinuous nature of the firing mechanism. Introducing the notion of an IF mode-locked state we are able to show that linear stability is consistent with the requirement of a negative IF Liapunov exponent. We apply our results to IF systems that may be used to study the entrainment of biological oscillators.