On Lyapunov Stability and Attractivity of Fractional Order Systems

Lyapunov stability is addressed here, which expands new knowledge of identifying continuous trajectories of fractional order systems that develops the ultimate goal to reach near or converge to its equilibria whenever one convincingly chooses the right Lyapunov functions. The notions of asymptotic stability, stability, and multi-order Mittag-Leffler stability were discussed for complicated nonlinear fractional order systems whenever associated different orders that may lie in $(0,1]$ and begin with the initial position posed at a random initial time take values on the real number line. The overview of this work is to give readers an enlightening insight into the so-called fractional Lyapunov direct method, which asserts how amazingly one can think of scalar Lyapunov functions to reasonably predict stability dynamics in large time, especially when time $t$ tends to $\infty$. We also establish some new sufficient conditions for stability and introduce a new notion of attractiveness of any bounded fixed solution or solution pairs that can be visualized in many such systems. The consequences of some results were adequate in exemplary models.