Effective Fokker-Planck Equation for Birhythmic Modified van der Pol Oscillator

We present an explicit solution based on the phase-amplitude approximation of the Fokker-Planck equation associated with the Langevin equation of the birhythmic modified van der Pol system. The solution enables us to derive probability distributions analytically as well as the activation energies associated to switching between the coexisting different attractors that characterize the birhythmic system. Comparing analytical and numerical results we find good agreement when the frequencies of both attractors are equal, while the predictions of the analytic estimates deteriorate when the two frequencies depart. Under the effect of noise the two states that characterize the birhythmic system can merge, inasmuch as the parameter plane of the birhythmic solutions is found to shrink when the noise intensity increases. The solution of the Fokker-Planck equation shows that in the birhythmic region, the two attractors are characterized by very different probabilities of finding the system in such a state. The probability becomes comparable only for a narrow range of the control parameters, thus the two limit cycles have properties in close analogy with the thermodynamic phases

Global stability analysis of birhythmicity in a self-sustained oscillator

We analyze global stability properties of birhythmicity in a self-sustained system with random excitations. The model is a multi-limit cycles variation of the van der Pol oscillatorintroduced to analyze enzymatic substrate reactions in brain waves. We show that the two frequencies are strongly influenced by the nonlinear coefficients $\alpha$ and $\beta$. With a random excitation, such as a Gaussian white noise, the attractor's global stability is measured by the mean escape time $\tau$ from one limit-cycle. An effective activation energy barrier is obtained by the slope of the linear part of the variation of the escape time $\tau$ versus the inverse noise-intensity 1/D. We find that the trapping barriers of the two frequencies can be very different, thus leaving the system on the same attractor for an overwhelming time. However, we also find that the system is nearly symmetric in a narrow range of the parameters.Comment: 17 pages, 8 figures, to appear on Choas, 201

Boundary Regional Controllability of Semilinear Systems Involving Caputo Time Fractional Derivatives

We study boundary regional controllability problems for a class of semilinear fractional systems. Sufficient conditions for regional boundary controllability are proved by assuming that the associated linear system is approximately regionally boundary controllable. The main result is obtained by using fractional powers of an operator and the fixed point technique under the approximate controllability of the corresponding linear system in a suitable subregion of the space domain. An algorithm is also proposed and some numerical simulations performed to illustrate the effectiveness of the obtained theoretical results.Comment: This is a preprint of a paper whose final and definite form is published in 'Libertas Mathematica (new series)', Volume 43 (2023), No. 1 [http://system.lm-ns.org/index.php/lm-ns/article/view/1488

Vegetative Propagation of Argania spinosa (L.) Skeels Cuttings: Effects of Nutrient Solution

The effects of the mineral composition of nutrient solution (Hoagland and Arnon (HA), Quoirin and Lepoivre (QL), Murashige and Skoog (MS), and Woody Plant Medium (WPM)), cutting type (softwood, semi-hardwood and hardwood) and cutting position (basal, medial, and apical) on sprouting and rooting performance of Argania spinosa cuttings were investigated. According to the results, the nutrient solution, cutting type and cutting position had an effect on the sprouting and adventitious rooting ability of A. spinosa cuttings. The leafy semi-hardwood cuttings taking from the basal positions and irrigated with Hoagland solution performed best and produced the highest number of roots (44.63), root length (28.86 cm), and had the highest rooting and survival percentage (63.81% and 96.09%, respectively). The nutrient solution applications caused a notable increase in sprouting and rooting potential of the argan tree. The highest values were recorded for HA and QL, while the MS and WPM gave the poorest result and the greatest mortality rate of cuttings. The cuttings type had also a pronounced effect on vegetative propagation of A. spinosa. The leafy semi-hardwood cuttings performed better than the leafy softwood cuttings, whereas leafless hardwood cuttings were completely unable to sprout and root even when treated with nutrient solutions. Thus, vegetative propagation of A. spinosa can best be achieved using basal leafy semi-hardwood cuttings irrigated with Hoagland nutrient solution

Boundary Controllability of Riemann-Liouville Fractional Semilinear Equations

We study the boundary regional controllability of a class of Riemann-Liouville fractional semilinear sub-diffusion systems with boundary Neumann conditions. The result is obtained by using semi-group theory, the fractional Hilbert uniqueness method, and Schauder's fixed point theorem. Conditions on the order of the derivative, internal region, and on the nonlinear part are obtained. Furthermore, we present appropriate sufficient conditions for the considered fractional system to be regionally controllable and, therefore, boundary regionally controllable. An example of a population density system with diffusion is given to illustrate the obtained theoretical results. Numerical simulations show that the proposed method provides satisfying results regarding two cases of the control operator.Comment: This is a preprint version of the paper published open access in 'Commun. Nonlinear Sci. Numer. Simul.' [https://doi.org/10.1016/j.cnsns.2023.107814]. Submitted 26/Jul/2022; Revised 08/Dec/2022 and 16/Oct/2023; Accepted for publication 30/Dec/2023; Available online 03/Jan/202

Regional Gradient Observability for Fractional Differential Equations with Caputo Time-Fractional Derivatives

We investigate the regional gradient observability of fractional sub-diffusion equations involving the Caputo derivative. The problem consists of describing a method to find and recover the initial gradient vector in the desired region, which is contained in the spacial domain. After giving necessary notions and definitions, we prove some useful characterizations for exact and approximate regional gradient observability. An example of a fractional system that is not (globally) gradient observable but it is regionally gradient observable is given, showing the importance of regional analysis. Our characterization of the notion of regional gradient observability is given for two types of strategic sensors. The recovery of the initial gradient is carried out using an expansion of the Hilbert Uniqueness Method. Two illustrative examples are given to show the application of the developed approach. The numerical simulations confirm that the proposed algorithm is effective in terms of the reconstruction error.Comment: This is a 22 pages preprint of a paper whose final and definite form is published in 'Int. J. Dyn. Control' (ISSN 2195-268X). Submitted 11/July/2022; Revised 07/Nov/22; and Accepted 26/Dec/202

Geophysical Characterization of Disturbances in the Phosphate Series of the OuladAbdoun, Morocco: Relationship with Atlasictectonics

The sedimentary phosphates series from the Upper Maastricht to the Lutetion of the OuladAbdoun sedimentary basin is almost identical at the basin-scale. It is made up by alternation of either phosphate or non-phosphate sub-horizontal levels. The series was characterized by rhythmic sedimentation under the form of successive elementary sequences. On the other hand, the series of SidiChennaneis distinguished by the presence of local disturbances defined by the miners under the term "dérangements". These "dérangements", which are less frequent at other mining sites (MEA Lahrech, El Hlassa, Point A), are a notable problem during extraction and remain a real obstacle at phosphate mining sites. The morphology of these "dérangements" is almost subcircular to subconical sinkholes and chaotic bodies of anarchical materials. They are fontis type paleokarsts, it is an amalgam of highly altered yellowish brecciated rusty material whose lithological nature reflects that of the surrounding series. They also reflect endokarst siliceous and ferruginous neoformedfacies in the empty spaces of the palaeokarst. The origin of the palaeokarstshasbeen linked to the presence of NE-SW trend faults that have favoured the alteration and dissolution of the gypsum and chalk facies of the Senonian. The regular spatial distribution of these fontisis well related to the regionalAtlasictectonics. This study aims to investigate these problems in its geological aspect, in order to characterize and understand their origin. The purpose of this work isto compare the results obtained by electric tomography, gravimetry and lineament mapping and match them with geological data to draw a meaningful conclusion on the existence of these disturbances and their spatial distribution in relation to tectonic

Analysis of a SIRI epidemic model with distributed delay and relapse

We investigate the global behaviour of a SIRI epidemic model with distributed delay and relapse. From the theory of functional differential equations with delay, we prove that the solution of the system is unique, bounded, and positive, for all time. The basic reproduction number $R_{0}$ for the model is computed. By means of the direct Lyapunov method and LaSalle invariance principle, we prove that the disease free equilibrium is globally asymptotically stable when $R_{0} < 1$. Moreover, we show that there is a unique endemic equilibrium, which is globally asymptotically stable, when $R_{0} > 1$.Comment: This is a preprint of a paper whose final and definite form is with 'Statistics Opt. Inform. Comput.', Vol. 7, No 2 (2019). See [http://www.iapress.org]. Submitted 26/July/2018; Revised and Accepted 22/Dec/201

Modified differentials and basic cohomology for Riemannian foliations

We define a new version of the exterior derivative on the basic forms of a Riemannian foliation to obtain a new form of basic cohomology that satisfies Poincar\'e duality in the transversally orientable case. We use this twisted basic cohomology to show relationships between curvature, tautness, and vanishing of the basic Euler characteristic and basic signature.Comment: 20 pages, references added, minor corrections mad

Application of Geophysics for the Detection of Derangement of Phosphate Layers in the Oulad Abdoun Basin in Morocco

The phosphate series of the basin of Oulad Abdoun begins in Maastrichtian with phosphate deposits relatively very marly. It ends at the Lutetian by a calcareous slab. Derangement is any disruption of the usual succession of the phosphate series and that which disrupts the evolution of the kinematic chain, leading to a decrease in production and profitability. In this case, we have a partially disturbed series and the disturbance consists of all the elements of the series (limestone, flint, marls and phosphate). The present work has been carried out in two ways: The present work has been carried out in two ways: The first one, purely geological, consists of the identification of the different layers of the Ouled Abdoun basin in the El Halassa site and their continuity to the outcrop. At the end of these observations, the basin shows derangement of two kinds: a disturbance on the scale of the whole series known as major disturbance, and a second which affects only part of the series. Thus, it is a minor or local disturbance. The second one, geophysics, is the application of three geophysical methods: electric tomography, magnetism, and refraction seismic. The correlation of these applications should result in delineating the mineralized zone and tracking all elements that in one way or another affect this mineralization. These elements are referred to as "derangement". The combination of the results of these two methods (vertical electrical survey and tomography) used allowed us to identify and map the disturbed places in the chosen area of El Halassa. The study will be extended to other sites and the results can be compared and correlated to understand the extent and origin of these disturbances