Fractional order spring/spring-pot/actuator element in a multibody system: Application of an expansion formula

Fractional order models of a spring/spring-pot and spring/spring-pot/actuator element connected into a multibody system are proposed in order to represent smart materials and components in adaptronic systems by introducing new tuning parameter. The models are introduced into dynamic equations via generalized forces and using the Lagrange's equations of the second kind in covariant form. Generalized forces are derived by taking into account fractional order derivatives in force–displacement relations and by using the principle of virtual work. The numerical scheme for solving fractional order differential equations proposed in Atanacković and Stanković (2008) is used in order to approximate fractional order derivative of a composite function appearing in the presented fractional order model. Numerical example for the multibody system with three degrees of freedom is presented. The results obtained for generalized forces are compared for different values of parameters in the fractional order derivative model.     

Uniform of four fractional-order nonlinear feedback synchronizations

This paper designs four fractional order nonlinear feedback synchronizations with the simple configuration, followed with their uniform. The closed system's stability is proved based on the fractional order stability theory. Resorted to the fractional order unified chaotic system, it is illustrated that the uniform includes the active, ordinary, dislocated, speed nonlinear feedback synchronizations and their mixed formulations. Numerical simulations show the effectiveness of the proposed methods.     

Fractional-order formulation of power-law and exponential distributions

We present a novel approach to data analysis using fractional order calculus. In principle the approach can be applied to any distribution and shows remarkable improvement even if the parameters of a particular distribution have been optimised to achieve the best fit to data. The method is demonstrated for two important distributions that are used in data analysis, namely, power-law and exponential distributions. We show that the approach can allow composite distributions to be constructed for improved accuracy and robustness in the characterisation of data sets.     

An analysis of the sharpness of monotonicity results via homotopy for sequential fractional operators

We investigate the connection between the sign of $\left({\Delta }_{1-\mu +a}^{\nu }{\Delta }_a^{\mu }f\right)\left(t\right)$ and the monotone behavior of $t\mapsto f\left(t\right)$. In particular, given a function $f:{\mathbb{N}}_a\to \mathbb{R}$ we consider the homotopy $G_{\nu }:\mathbb{R}\times [0,1]\to \mathbb{R}$ defined by $G_{\nu }\left(f\left(a\right),\eta \right)≔\eta \cdot \frac{1}{2}(1-\nu )\left(\nu \right)f\left(a\right)$, and we analyze the effect of changing $\eta \in [0,1]$ in the condition $\left({\Delta }_{1-\mu +a}^{\nu }{\Delta }_a^{\mu }f\right)\left(t\right)\ge G_{\nu }\left(f\left(a\right),\eta \right)$. We show that as $\eta \to 1^{-}$ there is an increasingly strong connection between the sign of $\left({\Delta }_{1-\mu +a}^{\nu }{\Delta }_a^{\mu }f\right)\left(t\right)$ and the monotone behavior of $f$.