Abstract
The purpose of this paper is to investigate to what extent cooperativity, that is, the absence of negative interactions, in Boolean networks with synchronous updating, imposes limits on chaos-like properties that are possible in such systems. Our focus is on notions of sensitive dependence on initial conditions, or a combination of sensitive dependence and large basins of attraction of exponentially long attractors, both of which are well-recognized hallmarks of chaotic dynamics in the Boolean context.We prove that a strong notion of sensitive dependence on initial conditions that formalizes decoherence along the attractor is precluded by cooperativity. Weaker notions of sensitive dependence that formalize decoherence at some time during the trajectory and sensitive dependence of the basin of attraction on initial conditions, respectively, are shown to be consistent with cooperativity, but if each regulatory function is binary AND or binary OR, in N-dimensional networks they impose an upper bound of ≈ 3 N on the lengths of attractors that can be reached from a fraction p≈1 of initial conditions. The upper bound is shown to be optimal. These results indicate that the transfer of analogous results for differential equations models crucially depends on the precise conceptualization of chaos in the Boolean context.MSC: 34C12, 39A33, 94C10.
Highlights
1 Introduction Many natural systems can be modeled with several types of dynamical systems, and it is of interest to study which properties of differential equations models carry over to certain types of difference equation models
Chaotic dynamics of Boolean networks is characterized by very long attractors, very few eventually frozen nodes, and high sensitivity to perturbations of initial conditions [ ]
6 Conclusion and future directions In this paper and its prequel [ ], we studied the problem whether cooperativity, that is, the total absence of negative interactions, precludes certain types of chaotic dynamics in Boolean networks with synchronous updating, at least under additional assumptions on the number of inputs and outputs per node
Summary
Many natural systems can be modeled with several types of dynamical systems, and it is of interest to study which properties of differential equations models carry over to certain types of difference equation models. Theorem Given any < p < and < c < , for all sufficiently large N , there exist p-cchaotic and p-unstable N -dimensional bi-quadratic cooperative Boolean networks Another hallmark of chaotic dynamics in Boolean networks is extensive damage propagation, which means that a small perturbation (such as a single-bit flip in an initial condition) tends to spread to a significant proportion of the nodes. Definition A Boolean network exhibits p-α-q-decoherence if with probability ≥ p a random one-bit flip s∗( ) in a randomly chosen initial condition s( ) results in trajectories with the property that for all sufficiently large t∗ > , the proportion of times t ∈ [ , t∗], for which the Hamming distance satisfies H(s(t), s∗(t)) ≥ αN , is at least q. It is straightforward to verify that conditions (i)-(iv) hold, with the all-important condition (iii) following from ( )
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