On Generalized Fractional Dynamical System With Order Lying in (0, 2): Stability Analysis, Chaotic Behaviour, Control and Synchronization

The generalized fractional dynamical system with order lying in (0, 2) is investigated. We present the stability analysis of that system using Mittag-Leffler function, the Gronwall-Bellman Lemma and Laplace transform. The bifurcation diagram of generalized fractional-order Chen system is given. We investigate a theorem to control the chaotic generalized fractional-order systems by linear feedback control. Two examples to achieve the theorem of control are given. The synchronization between two different chaotic generalized fractional systems is presented. We give a theorem to calculate the control functions which achieve synchronization. This theorem is applied to achieve the synchronization between different generalized fractional-order systems with order lying in (0, 1]. And, also, used to achieve the synchronization between the identical generalized fractional-order L\"{u} systems with order lying in [1, 2). There exist an agreement among analytical results and numerical treatments for stability, control and synchronization theorems.

Existence and stability results for Caputo fractional stochastic differential equations with Lévy noise

In this paper, the existence of solution of stochastic fractional differential equations with L?vy noise is established by the Picard-Lindel?f successive approximation scheme. The stability of nonlinear stochastic fractional dynamical system with L?vy noise is obtained using Mittag Leffler function. Examples are provided to illustrate the theory.

Controllability criteria of fractional differential dynamical systems with non-instantaneous impulses

This manuscript prospects the controllability criteria of non-instantaneous impulsive Volterra type fractional differential systems. By enroling an appropriate Gramian matrix that is often defined by the Mittag-Leffler function and with the assistance of Laplace transform, the necessary and sufficiency conditions for the controllability of non-instantaneous impulsive Volterra-type fractional differential equations are derived by using algebraic approach and Cayley–Hamilton theorem. An important feature present in our paper is that we have taken non-instantaneous impulses into the fractional order dynamical system and studied the controllability analysis, since this do not exist in the available source of literature. Inclusively, we have provided two illustrative examples with the existence of non-instantaneous impulse into the fractional dynamical system. So this demonstrates the validity and efficacy of our obtained criteria of the main section.

On the Solution of an Imprecisely Defined Nonlinear Time-Fractional Dynamical Model of Marriage

The present paper investigates the numerical solution of an imprecisely defined nonlinear coupled time-fractional dynamical model of marriage (FDMM). Uncertainties are assumed to exist in the dynamical system parameters, as well as in the initial conditions that are formulated by triangular normalized fuzzy sets. The corresponding fractional dynamical system has first been converted to an interval-based fuzzy nonlinear coupled system with the help of a single-parametric gamma-cut form. Further, the double-parametric form (DPF) of fuzzy numbers has been used to handle the uncertainty. The fractional reduced differential transform method (FRDTM) has been applied to this transformed DPF system for obtaining the approximate solution of the FDMM. Validation of this method was ensured by comparing it with other methods taking the gamma-cut as being equal to one.