Volume-preserving normal forms of Hopf-zero singularity

A practical method is described for computing the unique generator of the algebra of first integrals associated with a large class of Hopf-zero singularity. The set of all volume-preserving classical normal forms of this singularity is introduced via a Lie algebra description. This is a maximal vector space of classical normal forms with first integral; this is whence our approach works. Systems with a non-zero condition on their quadratic parts are considered. The algebra of all first integrals for any such system has a unique (modulo scalar multiplication) generator. The infinite level volume-preserving parametric normal forms of any non-degenerate perturbation within the Lie algebra of any such system is computed, where it can have rich dynamics. The associated unique generator of the algebra of first integrals are derived. The symmetry group of the infinite level normal forms are also discussed. Some necessary formulas are derived and applied to appropriately modified R\"{o}ssler and generalized Kuramoto--Sivashinsky equations to demonstrate the applicability of our theoretical results. An approach (introduced by Iooss and Lombardi) is applied to find an optimal truncation for the first level normal forms of these examples with exponentially small remainders. The numerically suggested radius of convergence (for the first integral) associated with a hypernormalization step is discussed for the truncated first level normal forms of the examples. This is achieved by an efficient implementation of the results using Maple

A geometric analysis of fast-slow models for stochastic gene expression

Stochastic models for gene expression frequently exhibit dynamics on several different scales. One potential time-scale separation is caused by significant differences in the lifetimes of mRNA and protein; the ratio of the two degradation rates gives a natural small parameter in the resulting chemical master equation, allowing for the application of perturbation techniques. Here, we develop a framework for the analysis of a family of ‘fast-slow’ models for gene expression that is based on geometric singular perturbation theory. We illustrate our approach by giving a complete characterisation of a standard two-stage model which assumes transcription, translation, and degradation to be first-order reactions. In particular, we present a systematic expansion procedure for the probability-generating function that can in principle be taken to any order in the perturbation parameter, allowing for an approximation of the corresponding propagator probabilities to that same order. For illustrative purposes, we perform this expansion explicitly to first order, both on the fast and the slow time-scales; then, we combine the resulting asymptotics into a composite fast-slow expansion that is uniformly valid in time. In the process, we extend, and prove rigorously, results previously obtained by Shahrezaei and Swain (Proc Natl Acad Sci USA 105(45):17256–17261, 2008) and Bokes et al. (J Math Biol 64(5):829–854, 2012; J Math Biol 65(3):493–520, 2012). We verify our asymptotics by numerical simulation, and we explore its practical applicability and the effects of a variation in the system parameters and the time-scale separation. Focussing on biologically relevant parameter regimes that induce translational bursting, as well as those in which mRNA is frequently transcribed, we find that the first-order correction can significantly improve the steady-state probability distribution. Similarly, in the time-dependent scenario, inclusion of the first-order fast asymptotics results in a uniform approximation for the propagator probabilities that is superior to the slow dynamics alone. Finally, we discuss the generalisation of our geometric framework to models for regulated gene expression that involve additional stages

Gevrey Solutions Of Singularly Perturbed Differential Equations

this article, we present an application where the existence of ireal" solutions can not be proved (at least at this time) without the Gevrey character of formal solutions. In the present form, our method does not provide optimal estimates for the type in the Gevrey estimates (i.e. the radius of convergence of the Borel transform) these estimates, though, are good enough to justify truncation at the least term. Optimal estimates have been achieved in the analog, but very special case of [29] by a judicious choice of the norms. Our method also does only prove local existence of solutions. Compare here with the results of Beno#t, Fruchard, Sch#fke and Wallet [5]: in the one dimensional case they obtain the existence of overstable solutions on ibig" domains (satisfying some geometrical condition) and good Gevrey estimates. We also would like to mention related work of Sibuya [32], Sch#fke [26] and the proof of an important conjecture on iturning points" of W. Wasow ([42], p.48) by C. Stenger [36] based on our results. 2 The equation