Based on recent studies by Guy Jumarie [1] which defines probability density of fractional order and fractional moments by using fractional calculus (fractional derivatives and fractional integration), this study expands the concept of probability density of fractional order by defining the fractional probability measure, which leads to a fractional probability theory parallel to the classical ...

‎in this paper, we investigate the chaotic behaviors of the fractional-order permanent magnet synchronous motor (pmsm) system. the necessary condition for the existence of chaos in the fractional-order pmsm system is deduced and an active controller is developed based on the stability theory for fractional systems. the presented control scheme  is simple and flexible, and it is suitable both fo...

A novel nonlinear fractional order sliding mode controller is proposed to control the chaotic atomic force microscope system in presence of uncertainties and disturbances. In the design of the suggested fractional order controller, conformable fractional order derivative is applied. The stability of the scheme is proved by means of the Lyapunov theory based on conformable fractional order deriv...

We develop the theory of hybrid fractional differential equations with the complex order $thetain mathbb{C}$, $theta=m+ialpha$, $0<mleq 1$, $alphain mathbb{R}$, in Caputo sense. Using Dhage's type fixed point theorem for the product of abstract nonlinear operators in Banach algebra; one of the operators is $mathfrak{D}$- Lipschitzian and the other one is completely continuous, we prove the exis...

this paper concerns the problem of robust stabilization of uncertain fractional-order non-autonomous systems. in this regard, a single input active control approach is proposed for control and stabilization of three-dimensional uncertain fractional-order systems. the robust controller is designed on the basis of fractional lyapunov stability theory. furthermore, the effects of model uncertainti...

‎In this paper, we investigate the chaotic behaviors of the fractional-order permanent magnet synchronous motor (PMSM) system. The necessary condition for the existence of chaos in the fractional-order PMSM system is deduced and an active controller is developed based on the stability theory for fractional systems. The presented control scheme  is simple and flexible, and it is suitable both fo...

  This paper concerns the problem of robust stabilization of uncertain fractional-order non-autonomous systems. In this regard, a single input active control approach is proposed for control and stabilization of three-dimensional uncertain fractional-order systems. The robust controller is designed on the basis of fractional Lyapunov stability theory. Furthermore, the effects of model uncertai...

In this paper, we concerned with positive solutions for higher order m-point nonlinear fractional boundary value problems with integral boundary conditions. We establish the criteria for the existence of at least one, two and three positive solutions for higher order m-point nonlinear fractional boundary value problems with integral boundary conditions by using a result from the theory of fixed...