The aim of this work is to describe the qualitative behavior of the solution set of a given system of fractional differential equations and limiting behavior of the dynamical system or flow defined by the system of fractional differential equations. In order to achieve this goal, it is first necessary to develop the Local theory for fractional nonlinear systems. This is done by the extension of the Local center manifold theorem, the stable manifold theorem and the Hartman-Grobman theorem to the scope of fractional differential systems. These latter two theorems establish that the qualitative behavior of the solution set of a nonlinear system of fractional differential equations near an equilibrium point is typically the same as the qualitative behavior of the solution set of the corresponding linearized system near the equilibrium point. Furthermore, we discuss the stability conditions for the equilibrium points of these systems. We point out that, the fractional derivative in these systems is in the Caputo sense.

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A mathematical model of a within-host viral infection is presented. A Local stability analysis of the model is conducted in two ways. At first, the basic reproduction number of the system is calculated. It is shown that when the reproduction number falls below unity, the disease free equilibrium (DFE) is globally asymptotically stable, and when it exceeds unity, the DFE is unstable and there exists a unique infectious equilibrium which may or may not be stable. In the case of instability, there exists an asymptotically stable periodic solution. Secondly, an analysis of Local center manifold shows that whenR0=1, a transcritical bifurcation occurs where upon increasing R0 greater than one the DFE loses stability and a Locally asymptotically positive infection equilibrium appears.

We prove that, for every complex Hilbert space H, every weak 2-Local derivation on B(H) or on k(H) is a linear derivation. We also establish that every weak 2-Local derivation on an atomic von Neumann algebra or on a compact C∗-algebra is a linear derivation.

In this paper, we investigate a discrete-time prey-predator model. The system is formulated by using the piecewise constant argument method for differential equations and taking into account Holling type III. The existence and Local behavior of equilibria are studied. We established that the system experienced both Neimark-Sacker and period-doubling bifurcations analytically by using bifurcation theory and the center manifold theorem. In order to control chaos and bifurcations, the state feedback method is implemented. Numerical simulations are also provided for the theoretical discussion.

The present paper deals with a toxin producing phytoplankton (TPP) -zooplankton interaction in spatial environment in the context of phytoplankton bloom. In the absence of diffusion the stability of the given system in terms of co-existence and hopf bifurcation has been discussed. After that TPP-zooplankton interaction is considered in spatiotemporal domain by assuming self diffusion in both population. It has been obtained that in the presence of diffusion given system becomes unstable (Turing instability) under certain conditions. Moreover, by applying the normal form theory and the center manifold reduction for partial differential equations (PDEs), the explicit algorithm determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions is derived. Finally, numerical simulations supporting the theoretical analysis are also included.

In this manuscript, a center manifold reduction of the flow of a non-hyperbolic equilibrium point on a planar dynamical system with the Caputo derivative is proposed. The stability of the non-hyperbolic equilibrium point is shown to be Locally asymptotically stable, under suitable conditions, by using the fractional Lyapunov direct method.