Synchronization of homoclinic chaos and implications for biological clocks

All artificial clocks are based on two-dimensional dynamical systems, thus on stable limit cycles according to the Poincaré - Bendixon theorem. Natural periodic phenomena in biology, ecology as well as economic (business, finance) cycles are ruled by multidimensional dynamics, which approaches sequentially the same saddle focus singularity. Thus, these clocks are characterized by homoclinic chaos of the Shilnikov type. It consists of regular orbits in phase space which repeat themselves with a very small spatial variance, but at erratic times. This geometric regularity makes it difficult to control chaos by exploiting geometric indicators. On the other hand, it is well known that very small periodic perturbations stabilize biological clocks (e.g. circadian rhythms, neuron synchronization in global perceptions, etc.). We conjecture that the close approach to a saddle focus yields a large susceptibility, thus making the homoclinic system proclive to undergo phase synchronization. We show evidence of this phenomenon in the case of a laser with feedback, and discuss the general problem of the metrology of natural clocks.