Zeno breaking, the Contact effect and sensitive behaviour in piecewise-linear systems

Non-smooth approximation to steep sigmoidal switching networks has proven to be a fruitful approach to analysis of their behaviour, for example in the study of dynamics of gene regulation. However, the introduction of discontinuities also leads to some analytic challenges, and in situations where the true system is believed to be smooth, it is especially important to be sure that the behaviour predicted by the non-smooth analysis is close to that of nearby smooth systems, and to interpret results accordingly. Here we use particular examples to highlight several challenges of this type, in which care needs to be taken in conducting or interpreting the non-smooth analysis. Autoregulation in piecewise linear models of gene networks leads to sliding motion in threshold hyperplanes or their intersections, and trajectories in which two (or more) variables spiral in to threshold intersections in finite time. In the latter case, an infinite number of threshold transitions occur during the finite time convergence to the intersection, as in Zeno's ``paradox''. After convergence, other variables continue to evolve (hence ``Zeno breaking''), and many possibilities arise, including non-uniqueness. Exotic forms of behavior through threshold regions (boundary layers) in the piecewise linear systems lead to interesting and sensitive behavior in nearby steep sigmoidal systems. Also, autoregulation in gene networks is actually accomplished by a sequence of steps, and introducing intermediate variables leads to more general piecewise linear systems with other kinds of non-uniqueness, such as what I call the "Contact" effect.