Boolean Dynamical Systems Research Articles

Necessary conditions for multistationarity in discrete dynamical systems

R. Thomas conjectured, 20 years ago, that the presence of a positive circuit in the interaction graph of a dynamical system is a necessary condition for the presence of several stable states. Recently, E. Remy et al. stated and proved the conjecture for Boolean dynamical systems. Using a similar approach, we generalize the result to discrete dynamical systems, and by focusing on the asynchronous dynamics that R. Thomas used in the course of his analysis of genetic networks, we obtain a more general variant of R. Thomas’ conjecture. In this way, we get a necessary condition for genetic networks to lead to differentiation.

Nonlinear reactive systems on a lattice viewed as Boolean dynamical systems

We present a stochastic, time-discrete Boolean model that mimics the mesoscopic dynamics of the desorption reactions A+A-->A+S and A+A-->S+S in a one-dimensional lattice. In the continuous-time limit, we derive a hierarchy of dynamical equations for the subset of moments involving contiguous lattice sites. The solution of the hierarchy allows to compute the exact dynamics of the mean coverage for both microscopic and coarse-grained initial conditions, which turn out to be different from the mean field predictions. The evolution equations for the mean coverage and the second-order moments are shown to be equivalent to those provided by a time-continuous master equation. The important role of higher-order fluctuations is brought out by the failure of a truncation scheme retaining only two-particle fluctuation correlations.

Solution of the Boolean Markus–Yamabe Problem

The main point in the design of content addressable memory would be under what conditions the state possessing the total information can attract all other states in the phase of the system. The problem can be formulated as a global asymptotic stability problem of Boolean dynamical systems. In this article we give a complete answer to this global asymptotic stability problem. The conditions employed involve the Hamming distance on the phase space {0,1}nas well as the spectral condition on the Jacobian Boolean matrix of F: {0,1}n→{0,1}n evaluated at each point of {0,1}n. This article furnishes a complete solution of the Boolean Markus–Yamabe problem.