In this paper the centers and limit cycles for a class of three-dimensional

cubic Kukles systems are investigated. First, by calculating and analyzing the common zeros of the first

ten singular point quantities, the necessary conditions for the origin being a center on

the center manifold are derived, and furthermore, the sufficiency of those conditions is proved using the Darboux

integrating method. Then, by calculating and analyzing the common zeros of the first three period

constants, the necessary and sufficient conditions for the origin being an isochronous

center on the center manifold are given. Finally, by proving the linear independence of

the first ten singular point quantities, it is demonstrated that the system can bifurcate ten

small-amplitude limit cycles near the origin under a suitable perturbation, which is a new lower bound for the number of limit cycles around a weak focus in a

three-dimensional cubic system.

In this paper we study the boundedness and unboundedness of the solutions of the smooth and discontinuous (SD) oscillator

\begin{equation*}

x''+f(x)x'+x-\frac{x}{\sqrt{x^{2}+\alpha^{2}}}=p(t).

\end{equation*}

Since $f(x)\neq 0$, the system is non-Hamiltonian, so we have to introduce some reversibility assumptions to apply a suitable twist theorem, for reversible maps with small twist. Moreover, when the nonnegative parameter $\alpha$ decreases to 0, the system becomes discontinuous. In this case, we need to introduce some suitable transformations to overcome the lack of regularity. We will prove that for any nonnegative parameter $\alpha$, when $p(t)$ is an odd periodic function satisfying $\left|\int^{2\pi}_{0}p(t)\sin t\,\dif\,t\right|<4$, all the solutions are bounded; when $p(t)$ satisfies $\left|\int^{2\pi}_{0}p(t)\sin t\,\dif\,t\right|>4$, the SD oscillator has unbounded solutions, and when $p(t)$ satisfies $\left|\int^{2\pi}_{0}p(t)\sin t\,\dif\,t\right|\geqslant 4+|F|_{\infty}$, all the solutions are unbounded.

This paper investigates the stability and bifurcation phenomena of a cholera transmission model in which individuals who have recovered from the disease may become susceptible again. The threshold for determining disease prevalence is established, and the parameter conditions for

the existence of equilibria are discussed. The Routh-Hurwitz criterion is applied to demonstrate the local asymptotic stability of equilibria.

By utilizing composite matrices and geometric techniques, the global dynamic behavior of the endemic equilibrium is investigated, and the sufficient conditions for its global asymptotic stability are derived. Furthermore, the disease-free equilibrium is a saddle-node when the basic reproductive number is 1, and tthe transcritical bifurcation in this case is discussed.