Stability analysis of fractional difference equations with delay.

Long-term memory is a feature observed in systems ranging from neural networks to epidemiological models. The memory in such systems is usually modeled by the time delay. Furthermore, the nonlocal operators, such as the "fractional order difference," can also have a long-time memory. Therefore, the fractional difference equations with delay are an appropriate model in a range of systems. Even so, there are not many detailed studies available related to the stability analysis of fractional order systems with delay. In this work, we derive the stability conditions for linear fractional difference equations with an arbitrary delay τ and even for systems with distributed delay. We carry out a detailed stability analysis for the cases of single delay with τ=1 and τ=2. The results are extended to nonlinear maps. The formalism can be easily extended to multiple time delays.

Study of low-dimensional nonlinear fractional difference equations of complex order.

We study the fractional maps of complex order, α e, for 0 < α < 1 and 0 ≤ r < 1 in one and two dimensions. In two dimensions, we study Hénon, Duffing, and Lozi maps, and in 1 d, we study logistic, tent, Gauss, circle, and Bernoulli maps. The generalization in 2 d can be done in two different ways, which are not equivalent for fractional order and lead to different bifurcation diagrams. We observed that the smooth maps, such as logistic, Gauss, Duffing, and Hénon maps, do not show chaos, while discontinuous maps, such as Bernoulli and circle maps,show chaos. The tent and Lozi map are continuous but not differentiable, and they show chaos as well. In 2 d, we find that the complex fractional-order maps that show chaos also show multistability. Thus, it can be inferred that the smooth maps of complex fractional order tend to show more regular behavior than the discontinuous or non-differentiable maps.

Singular points in the solution trajectories of fractional order dynamical systems.

Dynamical systems involving non-local derivative operators are of great importance in Mathematical analysis and applications. This article deals with the dynamics of fractional order systems involving Caputo derivatives. We take a review of the solutions of linear dynamical systems 0 C D t α X ( t ) = A X ( t ) , where the coefficient matrix A is in canonical form. We describe exact solutions for all the cases of canonical forms and sketch phase portraits of planar systems. We discuss the behavior of the trajectories when the eigenvalues λ of 2 × 2 matrix A are at the boundary of stable region, i.e., | a r g ( λ ) | = α π 2 . Furthermore, we discuss the existence of singular points in the trajectories of such planar systems in a region of C , viz. Region II. It is conjectured that there exists a singular point in the solution trajectories if and only if λ ∈ Region II.

Stability and bifurcation analysis of a generalized scalar delay differential equation.

This paper deals with the stability and bifurcation analysis of a general form of equation D(α)x(t)=g(x(t),x(t-τ)) involving the derivative of order α ∈ (0, 1] and a constant delay τ ≥ 0. The stability of equilibrium points is presented in terms of the stability regions and critical surfaces. We provide a necessary condition to exist chaos in the system also. A wide range of delay differential equations involving a constant delay can be analyzed using the results proposed in this paper. The illustrative examples are provided to explain the theory.