Fractional dynamic system simulating the growth of microbe

There are different approaches that indicate the dynamic of the growth of microbe. In this research, we simulate the growth by utilizing the concept of fractional calculus. We investigate a fractional system of integro-differential equations, which covers the subtleties of the diffusion between infected and asymptomatic cases. The suggested system is applicable to distinguish the presentation of growth level of the infection and to approve if its mechanism is positively active. An optimal solution under simulation mapping assets is considered. The estimated numerical solution is indicated by employing the fractional Tutte polynomials. Our methodology is based on the Atangana–Baleanu calculus (ABC). We assess the recommended system by utilizing real data.

Study of Chaotic and Regular Modes of the Fractional Dynamic System of Selkov

The paper investigates the dynamic modes of the Sel’kov fractional self-oscillating system in order to simulate the interaction of cracks. The spectra of the maximum Lyapunov exponents, constructed depending on the parameters of the dynamic system, are used as a research tool. The maximum Lyapunov exponents were constructed according to the Benettin-Wolf algorithm. It is shown that the existence of chaotic regimes is possible. In particular, the spectrum of the maximum Lyapunov exponents of the order of the fractional derivative contains positive values, which indicates the presence of a chaotic regime. Phase trajectories were also constructed to confirm these results. It was also confirmed that the orders of fractional derivatives are responsible for dissipation in the system under consideration.

Existence, stability and controllability results of fractional dynamic system on time scales with application to population dynamics

In this manuscript, we investigate the existence, uniqueness, Hyer-Ulam stability and controllability analysis for a fractional dynamic system on time scales. Mainly, this manuscript has three segments: In the first segment, we give the existence of solutions. The second segment is devoted to the study of stability analysis while in the last segment, we establish the controllability results. We use the Banach and nonlinear alternative Lery-Schauder–type fixed point theorem to establish these results. Also, we give some numerical examples for different time scales. Moreover, we give two applications to outline the effectiveness of these obtained results.

Modeling of fracture concentration by Sel’kov fractional dynamic system

Microseismic phenomena are studied by a Sel’kov generalized nonlinear dynamic system. This system is mainly applied in biology to describe substrate and product glycolytic oscillations. Thus, Sel’kov dynamic system can also describe interaction of two types of fractures in an elastic-friable medium. The first type includes seed fractures with lower energy and the second type are large fractures which generate microseisms. The first type of fractures are triggers for the second type of fractures. However opposite transition is possible. For example, when large fractures lose their energy and partially become seed ones. After their concentration increase, the process repeats providing auto oscillation character of microseism sources. Generalization of Sel’kov dynamic system is its analogue which is based on hereditarity. Hereditarity is studied within hereditary mechanics and it shows that a dynamic system can “remember” for some time the impact which was made upon it. It is typical for viscoelastic and yielding mediums. The Sel’kov generalized dynamic system will be called Sel’kov fractional dynamic system as long as from the point of view of mathematical description, it can be represented in the form of a system of differential equations with fractional derivatives. Fractional derivative orders are associated with system hereditarity and are responsible for energy dissipation intensity emitted by first- and second-type fractures. In the paper, the Sel’kov fractional dynamic model was numerically solved by Adams-Bashforth-Moulton method. Oscillo-grams and phase trajectories were plotted. It was shown that fractional dynamic model may have relaxation and damped oscillations.