We introduce simple models of genetic regulatory networks and we proceed to the mathematical analysis of their dynamics. The models are discrete time dynamical systems generated by piecewise affine contracting mappings whose variables represent gene expression levels. These models reduce to boolean networks in one limiting case of a parameter, and their asymptotic dynamics approaches that of a differential equation in another limiting case of this parameter. For intermediate values, the model present an original phenomenology which is argued to be due to delay effects. This phenomenology is not limited to piecewise affine model but extends to smooth nonlinear discrete time models of regulatory networks. In a first step, our analysis concerns general properties of networks on arbitrary graphs (characterisation of the attractor, symbolic dynamics, Lyapunov stability, structural stability, symmetries, etc). In a second step, focus is made on simple circuits for which the attractor and its changes with parameters are described. In the negative circuit of 2 genes, a thorough study is presented which concern stable (quasi-)periodic oscillations governed by rotations on the unit circle - with a rotation number depending continuously and monotonically on threshold parameters. These regular oscillations exist in negative circuits with arbitrary number of genes where they are most likely to be observed in genetic systems with non-negligible delay effects.

We study the asymptotic dynamics of maps which are piecewise contracting on a compact space. These maps are Lipschitz continuous, with Lipschitz constant smaller than one, when restricted to any piece of a finite and dense union of disjoint open pieces. We focus on the topological and the dynamical properties of the (global) attractor of the orbits that remain in this union. As a starting point, we show that the attractor consists of a finite set of periodic points when it does not intersect the boundary of a contraction piece, which complements similar results proved for more specific classes of piecewise contracting maps. Then, we explore the case where the attractor intersects these boundaries by providing examples that show the rich phenomenology of these systems. Due to the discontinuities, the asymptotic behaviour is not always properly represented by the dynamics in the attractor. Hence, we introduce generalized orbits to describe the asymptotic dynamics and its recurrence and transitivity properties. Our examples include transitive and recurrent attractors, that are either finite, countable, or a disjoint union of a Cantor set and a countable set. We also show that the attractor of a piecewise contracting map is usually a Lebesgue measure-zero set, and we give conditions ensuring that it is totally disconnected. Finally, we provide an example of piecewise contracting map with positive topological entropy and whose attractor is an interval.

We study the complexity of the itineraries of injective piecewise contracting maps on the interval. We prove that for any such map the complexity function of any itinerary is eventually affine. We also prove that the growth rate of the complexity is bounded from above by the number N − 1 of discontinuities of the map. To show that this bound is optimal, we construct piecewise affine contracting maps whose itineraries all have the complexity (N − 1)n + 1. In these examples, the asymptotic dynamics takes place in a minimal Cantor set containing all the discontinuities.

We consider p(n) the number of partitions of a natural number n, starting from an expression derived by L. Báez-Duarte in [1] by relating its generating function f (t) with the characteristic functions of a family of sums of independent random variables indexed by t. The asymptotic formula for p(n) follows then from a local central limit theorem as t ↑ 1 suitably with n → ∞. We take further that analysis and compute formulae for the terms that compose that expression, which accurately approximate them as t ↑ 1. Those include the generating function f and the cumulants of the random variables. After developing an asymptotic series expansion for the integral term we obtain an expansion for p(n) that can be simplified as follows: for each N > 0,The coefficients D ℓ are positive and have simple expressions as finite sums of combinatorial numbers, r n = 2π 2 3 (n − 1 24 ) + 1 4 and the remainder satisfies n N/2 R N +1 → 0 as n → ∞.The cumulants are given by series of rational functions and the approximate formulae obtained could be also of independent interest in other contexts.

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