Generating Multidirectional Variable Hidden Attractors via Newly Commensurate and Incommensurate Non-Equilibrium Fractional-Order Chaotic Systems

This article investigates a non-equilibrium chaotic system in view of commensurate and incommensurate fractional orders and with only one signum function. By varying some values of the fractional-order derivative together with some parameter values of the proposed system, different dynamical behaviors of the system are explored and discussed via several numerical simulations. This system displays complex hidden dynamics such as inversion property, chaotic bursting oscillation, multistabilty, and coexisting attractors. Besides, by means of adapting certain controlled constants, it is shown that this system possesses a three-variable offset boosting system. In conformity with the performed simulations, it also turns out that the resultant hidden attractors can be distributively ordered in a grid of three dimensions, a lattice of two dimensions, a line of one dimension, and even arbitrariness in the phase space. Through considering the Caputo fractional-order operator in all performed simulations, phase portraits in two- and three-dimensional projections, Lyapunov exponents, and the bifurcation diagrams are numerically reported in this work as beneficial exit results.     

A class of degenerate fractional evolution systems in banach spaces

By using the notion of strongly (B, p)-sectorial operator and fractional differential calculus, we analyze the unique solvability of the Cauchy and Showalter problems for a class of degenerate fractional evolution systems. The results are used for the analysis of partial differential equations of fractional order with respect to the time variable.     

Blowing‐up solutions to two‐times fractional differential equations

Nonexistence results for a class of two‐times differential equations with fractional derivatives of orders between zero and one are presented. Furthermore, the result is extended to a two‐times system of two differential equations with fractional derivatives of orders between zero and one.     

Optimal Solutions to Relaxation in Multiple Control Problems of Sobolev Type with Nonlocal Nonlinear Fractional Differential Equations

We introduce the optimality question to the relaxation in multiple control problems described by Sobolev-type nonlinear fractional differential equations with nonlocal control conditions in Banach spaces. Moreover, we consider the minimization problem of multi-integral functionals, with integrands that are not convex in the controls, of control systems with mixed nonconvex constraints on the controls. We prove, under appropriate conditions, that the relaxation problem admits optimal solutions. Furthermore, we show that those optimal solutions are in fact limits of minimizing sequences of systems with respect to the trajectory, multicontrols, and the functional in suitable topologies.     

Chaotic dynamics in a novel COVID-19 pandemic model described by commensurate and incommensurate fractional-order derivatives

Nonlinear Dynamics - Mathematical models based on fractional-order differential equations have recently gained interesting insights into epidemiological phenomena, by virtue of their memory effect...     

Optimal controls for second-order stochastic differential equations driven by mixed-fractional Brownian motion with impulses

We study optimal control problems for a class of second-order stochastic differential equation driven by mixed-fractional Brownian motion with non-instantaneous impulses. By using stochastic analysis theory, strongly continuous cosine family, and a fixed point approach, we establish the existence of mild solutions for the stochastic system. Moreover, the optimal control results are derived without uniqueness of mild solutions of the stochastic system. Finally, the main results are validated with the aid of an example.     

Three-dimensional reconstruction of aerogels from TEM images

The knowledge of the internal structure of porous materials is of main importance to compute their physical properties. This article focuses on base-catalyzed and colloidal silica aerogels, which are fractal materials and we use an original method for the reconstruction of these aerogels from TEM images. The method used is iterative and leads to the same fractal dimension as the real material, computed from the internal two-point correlation function. Unlike the reconstruction of porous materials found in literature, our method is based on the distribution of matter and not of the porous network, and has the additional advantage of being only half statistical, i.e. only the coordinates in the z direction are statistically obtained while the coordinates x and y of the elements of matter are accurately obtained from TEM images of our samples     

A time-fractional competition ecological model with cross-diffusion

This paper is concerned with some mathematical and numerical aspects of a Lotka-Volterra competition time-fractional reaction-diffusion system with cross-diffusion effects. First, we study the existence of weak solutions of the model following the well-known Faedo-Galerkin approximation method and convergence arguments. We demonstrate the convergence of approximate solutions to actual solutions using the energy estimates. Next, the Galerkin finite element scheme is proposed for the considered model. Further, various numerical simulations are performed to show that the fractional-order derivative plays a significant role on the morphological changes of the considered competition model.