[en] This paper studies the robust adaptive control of fractional-order chaotic systems with system uncertainties and bounded external disturbances. Based on a proposed lemma, quadratic Lyapunov functions are used in the stability analysis and fractional-order adaptation laws are designed to update the controller parameters. By employing the fractional-order expansion of classical Lyapunov stability method, a robust controller is designed for fractional-order chaotic systems. The system states asymptotically converge to the origin and all signals in the closed-loop system remain bounded. A counterexample is constructed to show that the fractional-order derivative of a function is less than zero does not mean that the function monotonically decreases (this property appears in many references). Finally, simulation results are presented to confirm our theoretical results.

[en] The aim of this paper is using an elementary method and the properties of the Bernoulli polynomials to establish a close relationship between the Euler numbers of the second kind $E_n^{\ast <!--\; ???\; -->}$ and the Dirichlet L-function $L(s,\mathrm{\chi <!--\; ??\; -->})$. At the same time, we also prove a new congruence for the Euler numbers $E_n$. That is, for any prime $p\equiv <!--\; ???\; -->1mod8$, we have $E_{\frac{p-<!--\; ???\; -->3}{2}}\equiv <!--\; ???\; -->0modp$. As an application of our result, we give a new recursive formula for one kind of Dirichlet L-functions.

[en] Rossby waves, belonging to the most important waves in the atmosphere and ocean, can affect the energy transfer of the atmosphere and ocean and have significant theoretical meaning and research value. In previous research performed with the theory and calculation method limit, the dissipation effect was commonly ignored. However, under the conditions of the weak linear approximation, the magnitude difference between nonlinear and dissipation is small, and the dissipation effect must be considered. In this paper, based on the classic Lie group approach, the $(3+1)$-dimensional quasi-geodetiophic vorticity equation with dissipation effect is solved. With the help of the solutions, we can better comprehend the influence of the dissipation effect on the propagation of Rossby waves.

[en] In this paper, we consider the Galerkin finite element approximations of the initial value problem for the nonlinear fractional stochastic partial differential equations with multiplicative noise. We study a spatial semidiscrete scheme with the standard Galerkin finite element method and a fully discrete scheme based on the Goreno–Mainardi–Moretti–Paradisi (GMMP) scheme. We establish strong convergence error estimates for both semidiscrete and fully discrete schemes.

[en] A new kind of multiple stochastic optimal stopping problem is formulated and its associated recursive variational inequalities are derived. We show that these variational inequalities can be solved exactly in a cascading manner. The relevance of the present problem in analyzing animal migration, which is an ecologically important problem, is also briefly discussed.

[en] In this paper, we explore dynamic properties of a stochastic cooperation-competition model. Some sufficient conditions are established for stochastic persistence and stochastic extinction of species. Studies suggest that the noise may have a positive effect on the persistence of species. We also analyze global asymptotic stability of positive solutions and give a stationary distribution of this stochastic model which has the ergodic property. Finally, some numerical simulations are presented to illustrate or complement our mathematical findings.

[en] In this paper, we review indispensable properties and characterizations of almost periodic functions and asymptotically almost periodic functions in Banach spaces. Special accent is put on the Stepanov generalizations of almost periodic functions and asymptotically almost periodic functions. We also recollect some basic results regarding equi-Weyl-almost periodic functions and Weyl-almost periodic functions. The class of asymptotically Weyl-almost periodic functions, introduced in this work, seems to be not considered elsewhere even in the scalar-valued case. We actually introduce eight new classes of asymptotically almost periodic functions and analyze relations between them. In order to make a picture as complete and clear as possible, several illustrating examples and counter-examples are given. It is worth noting that the topics dealt with in this paper seem to be of an intrinsic connection with the problem of existence and uniqueness of solutions of differential and difference equations, in both determinist and stochastic cases.

[en] In this study, a new discrete SI epidemic model is proposed and established from SI fractional-order epidemic model. The existence conditions, the stability of the equilibrium points and the occurrence of bifurcation are analyzed. By using the center manifold theorem and bifurcation theory, it is shown that the model undergoes flip and Neimark–Sacker bifurcation. The effects of step size and fractional-order parameters on the dynamics of the model are studied. The bifurcation analysis is also conducted and our numerical results are in agreement with theoretical results.

[en] There has been an increasing interest in studying fractional-order chaotic systems and their synchronization. In this paper, the fractional-order form of a system with stable equilibrium is introduced. It is interesting that such a three-dimensional fractional system can exhibit chaotic attractors. Full-state hybrid projective synchronization scheme and inverse full-state hybrid projective synchronization scheme have been designed to synchronize the three-dimensional fractional system with different four-dimensional fractional systems. Numerical examples have verified the proposed synchronization schemes.

[en] The present article investigates the effects of diffusion-thermo, thermal radiation, and magnetic field of strength $B_0$ on the time dependent MHD flow of Jeffrey nanofluid past over a porous medium in a rotating frame. The plate is assumed vertically upward along the x-axis under the effect of cosine oscillation. Silver nanoparticles are uniformly dispersed into engine oil, which is taken as a base fluid. The equations which govern the flow are transformed into a time fractional model using Atangana–Baleanu time fractional derivative. To obtain exact expressions for velocity, temperature, and concentration profiles, the Laplace transform technique, along with physical initial and boundary conditions, is used. The behaviors of the fluid flow under the impact of corresponding dimensionless parameters are shown graphically. The variations in Nusselt number and Sherwood number of relative parameters are found numerically and shown in tabular form. It is worth noting that the rate of heat transfer of engine oil is enhanced by 15.04% when the values of volume fraction of silver nanoparticles vary from 0.00 to 0.04, as a result the lubricant properties are improved.