Positivity of Continuous-Time Descriptor Systems With Time Delays

This technical note is concerned with positivity characteristic of continuous-time descriptor systems with time delays. First, a set of necessary and sufficient conditions is presented to check the property. Then, considering a descriptor time-delay system with two assumptions, a new time-delay system is established whose positivity is equivalent to that of the original system. Furthermore, a set of necessary and sufficient conditions is provided to check the positivity of the new system. Finally, a numerical example is given to illustrate the validity of the results obtained

Stability of fractional order systems

The theory and applications of fractional calculus (FC) had a considerable progress during the last years. Dynamical systems and control are one of the most active areas, and several authors focused on the stability of fractional order systems. Nevertheless, due to the multitude of efforts in a short period of time, contributions are scattered along the literature, and it becomes difficult for researchers to have a complete and systematic picture of the present day knowledge. This paper is an attempt to overcome this situation by reviewing the state of the art and putting this topic in a systematic form. While the problem is formulated with rigour, from the mathematical point of view, the exposition intends to be easy to read by the applied researchers. Different types of systems are considered, namely, linear/nonlinear, positive, with delay, distributed, and continuous/discrete. Several possible routes of future progress that emerge are also tackled

Aberrant behaviours of reaction diffusion self-organisation models on growing domains in the presence of gene expression time delays

Turing’s pattern formation mechanism exhibits sensitivity to the details of the initial conditions suggesting that, in isolation, it cannot robustly generate pattern within noisy biological environments. Nonetheless, secondary aspects of developmental self-organisation, such as a growing domain, have been shown to ameliorate this aberrant model behaviour. Furthermore, while in-situ hybridisation reveals the presence of gene expression in developmental processes, the influence of such dynamics on Turing’s model has received limited attention. Here, we novelly focus on the Gierer–Meinhardt reaction diffusion system considering delays due the time taken for gene expression, while incorporating a number of different domain growth profiles to further explore the influence and interplay of domain growth and gene expression on Turing’s mechanism. We find extensive pathological model behaviour, exhibiting one or more of the following: temporal oscillations with no spatial structure, a failure of the Turing instability and an extreme sensitivity to the initial conditions, the growth profile and the duration of gene expression. This deviant behaviour is even more severe than observed in previous studies of Schnakenberg kinetics on exponentially growing domains in the presence of gene expression (Gaffney and Monk in Bull. Math. Biol. 68:99–130, 2006). Our results emphasise that gene expression dynamics induce unrealistic behaviour in Turing’s model for multiple choices of kinetics and thus such aberrant modelling predictions are likely to be generic. They also highlight that domain growth can no longer ameliorate the excessive sensitivity of Turing’s mechanism in the presence of gene expression time delays. The above, extensive, pathologies suggest that, in the presence of gene expression, Turing’s mechanism would generally require a novel and extensive secondary mechanism to control reaction diffusion patterning

Asymptotic behaviour of mild solution of nonlinear stochastic partial functional equations

This paper presents conditions to assure existence, uniqueness and stability for impulsive neutral stochastic integrodifferential equations with delay driven by Rosenblatt process and Poisson jumps. The Banach fixed point theorem and the theory of resolvent operator developed by Grimmer [R.C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Am. Math. Soc., 273(1):333–349, 1982] are used. An example illustrates the potential benefits of these results

Non-Lyapunov stability of the fractional-order time-varying delay systems

U ovom radu, kriterijumi stabilnosti na konačnom vremenskom intervalu su prošireni na nelinearne nehomogene perturbovane sisteme necelobrojnog reda koji sadrže višestruka vremenski promenljiva kašnjenja. Dobijeni su dovoljni uslovi stabilnosti za sisteme necelog reda sa višestrukim vremenskim kašnjenjima korišćenjem generalizovanog i klasičnog Gronwallovog pristupa. Numerički primer je dat u cilju ilustracije značaja dobijenog rezultata.In this paper, the finite-time stability criteria are extended to nonlinear nonhomogeneous perturbed fractional-order systems including multiple time-varying delays. The sufficient conditions of a stability for the fractional systems with multiple time delays are obtained by using the generalized and classical Gronwall's approach. A numerical example is presented to illustrate the validity of the obtained result