Calculation of the Statistical Properties in Intermittency Using the Natural Invariant Density

We use the natural invariant density of the map and the Perron–Frobenius operator to analytically evaluate the statistical properties for chaotic intermittency. This study can be understood as an improvement of the previous ones because it does not introduce assumptions about the reinjection probability density function in the laminar interval or the map density at pre-reinjection points. To validate the new theoretical equations, we study a symmetric map and a non-symmetric one. The cusp map has symmetry about x=0, but the Manneville map has no symmetry. We carry out several comparisons between the theoretical equations here presented, the M function methodology, the classical theory of intermittency, and numerical data. The new theoretical equations show more accuracy than those calculated with other techniques.

A few problems connected with invariant measures of Markov maps - verification of some claims and opinions that circulate in the literature

It is well known that C2-transformation φ of the unit interval into itself with a Markov partition (2.1) π = {Ik : k ∈ K} admits φ-invariant density g (g ≥ 0, ∥g∥ = 1) if: (2.2) ∣(φn)′∣ ≥ C1 > 1 for some n (expanding condition); (2.3) ∣φ″(x)/(φ′(y))2∣ ≤ C2 < ∞ (second derivative condition); and (2.4) #π < ∞ or φ (Ik) = [0, 1], for each Ik ∈ π. If (2.4) is deleted, then the situation dramatically changes. The cause of this fact was elucidated in connection with so-called Adler’s Theorem ([1] and [2]). However after that time in the literature occur claims and opinions concerning the existence of invariant densities and their properties for Markov Maps, which satisfy (2.2), (2.3) and do not satisfy (2.4), revealing unacquaintance with that question. In this note we discuss the problems arising from the mentioned claims and opinions. Some solutions of that problems are given, in a systematic way, on the base of the already published results and by providing appropriate examples.

Dynamics of drainage under stochastic rainfall in river networks

We consider a linearized dynamical system modeling the flow rate of water along the rivers and hillslopes of an arbitrary watershed. The system is perturbed by a random rainfall in the form of a compound Poisson process. The model describes the evolution, at daily time scales, of an interconnected network of linear reservoirs and takes into account the differences in flow celerity between hillslopes and streams as well as their spatial variation. The resulting stochastic process is a piece-wise deterministic Markov process of the Orstein–Uhlembeck type. We provide an explicit formula for the Laplace transform of the invariant density of streamflow in terms of the geophysical parameters of the river network and the statistical properties of the precipitation field. As an application, we include novel formulas for the invariant moments of the streamflow at the watershed’s outlet, as well as the asymptotic behavior of extreme discharge events, and conditions for the statistical scaling of streamflow with respect to Horton order.

Probability Analysis of a Perturbed Epidemic System with Relapse and Cure

This paper is devoted to a continuous-time stochastic differential system which is derived by incorporating white noise to a deterministic [Formula: see text] epidemic model with mass action incidence, cure and relapse. We focus on the impact of a relapse on the asymptotic properties of the stochastic system. We show that the relapse encourages the persistence of the disease in the population and we determine the threshold of the relapse rate, above the threshold the disease prevails in the population. Furthermore, we show that there exists a unique density function of solutions which converges in [Formula: see text], under certain conditions of the parameters to an invariant density.