arXiv subjects
Publications and source records attributed to Alexandre M. Alves.
In this paper, we present two new results to the classical Floquet theory, which provides the Floquet multipliers for two classes of the planar periodic system. One these results provides the Floquet multipliers independently of the solution of system. To demonstrate the application of these analytical results, we consider a cholera epidemic model with phage dynamics and seasonality incorporated.
In this paper we consider Abel equation $x' = g(t)x^2+f(t)x^3$, where $f$ and $g$ are analytical functions. We proved that if the equation has a center at $x=0$, then the Moment Conditions, i. e., $m_k=\int_{-1}^1f(t)(G(t))^kdt=0,~~k=0,1,2$, is satisfied where $G(t)=\int_{-1}^tg(s)ds$. Besides, we give partial a positive answer to a conjecture proposed by Y. Lijun and T. Yun in 2001.
Abel equations of the form $x'(t)=f(t)x^3(t)+g(t)x^2(t)$, $t \in [-a,a]$, where $a>0$ is a constant, $f$ and $g$ are continuous functions, are of interest because of their close relation to planar vector fields. If $f$ and $g$ are odd functions, we prove, in this paper, that the Abel equation has a center at the origin. We also consider a class of polynomial differential equations $\dot{x} = -y+P_n(x,y)$ and $\dot{y} = x+Q_n(x,y)$, where $P_n$ and $Q_n$ are homogeneous polynomials of degree $n$. Using the results obtained for Abel's equation, we obtain a new subclass of systems having a center at the origin.