Stability and ψ-algebraic decay of the solution to ψ-fractional differential system

In this article, we focus on stability and ψ-algebraic decay (algebraic decay in the sense of ψ-function) of the equilibrium to the nonlinear ψ-fractional ordinary differential system. Before studying the nonlinear case, we show the stability and decay for linear system in more detail. Then we establish the linearization theorem for the nonlinear system near the equilibrium and further determine the stability and decay rate of the equilibrium. Such discussions include two cases, one with ψ-Caputo fractional derivative, another with ψ-Riemann–Liouville derivative, where the latter is a bit more complex than the former. Besides, the integral transforms are also provided for future studies.

Riemannian metrics on Lie groupoids

We introduce a notion of metric on a Lie groupoid, compatible with multiplication, and we study its properties. We show that many families of Lie groupoids admit such metrics, including the important class of proper Lie groupoids. The exponential map of these metrics allows us to establish a linearization theorem for Riemannian groupoids, obtaining both a simpler proof and a stronger version of the Weinstein–Zung linearization theorem for proper Lie groupoids. This new notion of metric has a simplicial nature which will be explored in future papers of this series.

A Fractional-Order Epidemic Model for Bovine Babesiosis Disease and Tick Populations

This paper shows that the epidemic model, previously proposed under ordinary differential equation theory, can be generalized to fractional order on a consistent framework of biological behavior. The domain set for the model in which all variables are restricted is established. Moreover, the existence and stability of equilibrium points are studied. We present the proof that endemic equilibrium point when reproduction numberR0>1is locally asymptotically stable. This result is achieved using the linearization theorem for fractional differential equations. The global asymptotic stability of disease-free point, whenR0<1, is also proven by comparison theory for fractional differential equations. The numeric simulations for different scenarios are carried out and data obtained are in good agreement with theoretical results, showing important insight about the use of the fractional coupled differential equations set to model babesiosis disease and tick populations.

Linearization of Impulsive Differential Equations with Ordinary Dichotomy

This paper presents a linearization theorem for the impulsive differential equations when the linear system has ordinary dichotomy. We prove that when the linear impulsive system has ordinary dichotomy, the nonlinear systemx˙(t)=A(t)x(t)+f(t,x),t≠tk,Δx(tk)=A~(tk)x(tk)+f~(tk,x),k∈ℤ, is topologically conjugated tox˙(t)=A(t)x(t),t≠tk,Δx(tk)=A~(tk)x(tk),k∈ℤ, whereΔx(tk)=x(tk+)-x(tk-),x(tk-)=x(tk), represents the jump of the solutionx(t)att=tk. Finally, two examples are given to show the feasibility of our results.