Fractional Stability

A fractional generalization of variations is used to define a stability of non-integer order. Fractional variational derivatives are suggested to describe the properties of dynamical systems at fractional perturbations. We formulate stability with respect to motion changes at fractional changes of variables. Note that dynamical systems, which are unstable ”in sense of Lyapunov”, can be stable with respect to fractional variations. The theory of integrals and derivatives of non-integer order goes back to Leibniz, Liouville, Riemann, Grunwald, and Letnikov [1, 2]. Fractional analysis has found many applications in recent studies in mechanics and physics. The interest in fractional integrals and derivatives has been growing continuously during the last few years because of numerous applications. In a short period of time the list of applications has been expanding. For example, it includes the chaotic dynamics [3, 4], material sciences [5, 6, 7], mechanics of fractal and complex media [8, 9], quantum mechanics [10, 11], physical kinetics [3, 12, 13, 14, 15], long-range dissipation [16, 17], non-Hamiltonian mechanics [18, 19], long-range interaction [22, 23, 24]. In this preprint, we use the fractional generalization of variation derivatives that are suggested in [19]. Fractional integrals and derivatives are used for stability problems (see for example [25, 26, 27, 28, 29]). We consider the properties of dynamical systems with respect to fractional variations [19]. We formulate stability with respect to motion changes at fractional changes of variables (see [20] pages 294-296). Note that dynamical systems, which are unstable ”in sense of Lyapunov”, can be stable with respect to fractional variations. Let us consider dynamical system that is defined by the ordinary differential equations. Suppose that the motion of dynamical system is described by the equations d dt yk = Fk(y), k = 1, ..., n. (1) Here y1, ..., yn be real variables that define the state of dynamical system. Let us consider the variation δyk of variables yk. The unperturbed motion is satisfied to zero value of variation δyk = 0. The variation describes that as function f(y) changes at changes of argument y. The first variation describes changes of function with respect to the first power of changes of y: δf(y) = D yf(y)dy, (2)