Fractional Differential Equations as Nonlocal Dynamical Systems: A new Perspective

This study [Cong, N. D., & Tuan, H. T. (2017) "Generation of nonlocal fractional dynamical systems by fractional differential equations" examines nonlinear Caputo fractional-order differential equations D^α x = f(t,x) for initial conditions x₀ in the reals R^d of dimension d and the extent to which this differential equation generates a nonlocal dynamical system on R^d . Our main conclusion is that this is generally not possible for dimensions, d, larger than one, as different trajectories may meet in finite time.

On the other hand, linear time-periodic systems, D^α x = A(t) x with A(t) = A(t+T), and period time T are yet to be rigorously analyzed in the fractional-order case. In the case of functional differential equations in the sense of Hale, the respective generated dynamical system on the space of continuous functions C plays an important role in the development of Floquet theory for linear time-periodic systems. Both functional and fractional differential equations show nonlocal behavior.

Thus, we study a framework of Caputo-type differential equations for the lower bound - ∞, namely D_-∞^α x = f(t,x), where an initial condition φ ∈ S must be prescribed to formulate an initial value problem and S is a subset of the space of continuous functions C(- ∞,0]. We show that this initial condition manifests as a time-dependent forcing term |F φ (t)| ≤ b (t+ η)^-α on the right-hand side of the differential equation, decaying algebraically, where b, η > 0. Furthermore, we use this bound to analyze solutions of linear time-autonomous systems.

Finally, we investigate how such a system may be proposed to generate a nonlocal dynamical system on S. In particular, we provide nontrivial conditions under which a solution trajectory could belong to S.