Exponential transients in continuous-time Liapunov systems

AbstractWe consider the convergence behavior of a class of continuous-time dynamical systems corresponding to so-called symmetric Hopfield nets studied in neural networks theory. We prove that such systems may have transient times that are exponential in the system dimension (i.e. number of “neurons”), despite the fact that their dynamics are controlled by Liapunov functions. This result stands in contrast to many proposed uses of such systems in, e.g. combinatorial optimization applications, in which it is often implicitly assumed that their convergence is rapid. An additional interesting observation is that our example of an exponential-transient continuous-time system (a simulated binary counter) in fact converges more slowly than any discrete-time Hopfield system of the same representation size. This suggests that continuous-time systems may be worth investigating for gains in descriptional efficiency as compared to their discrete-time counterparts