Chaotic Dynamics of Autonomous Boolean Networks

Abstract

In this chapter, I apply autonomous Boolean networks to one particularly popular topic of complex systems research: deterministic chaos. I first introduce the concept of deterministic chaos in Sect. 4.2 and then extend in Sect. 4.3 the previous work on Boolean chaos with a simple autonomous Boolean network that has been proposed in a similar form in an early theoretical study by Ghil and Mullhaupt (J Stat Phys 41: 125, 1985). I measure and analyze the dynamics of an experimental implementation and develop a time delay piecewise-linear switching model for autonomous Boolean networks in Sect. 4.4. Both experimental and simulated dynamics agree qualitatively in power spectrum and autocorrelation. The main result of this chapter is the development of a particularly simple autonomous Boolean network for generating Boolean chaos using guidelines from Boolean network models. These guidelines, however, cannot predict with certainty whether a network will display complex dynamics. For example, a prediction of complex dynamics with a Boolean network model breaks down when the network is realized in the experiment, where the network instead shows transient relaxation towards a fixed point (Results of this chapter are published in reference Rosin et al. Chaos 23: 025102, 2013.).