### A contribution to the theory of stability of differential equations in Banach space

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In the theory of autonomous perturbations of periodic solutions of ordinary differential equations the method of the Poincaré mapping has been widely used. For the analysis of properties of this mapping in the case of two-dimensional systems, a result first obtained probably by Diliberto in 1950 is sometimes used. In the paper, this result is (partially) extended to a certain class of autonomous ordinary differential equations of higher dimension.

Under suitable hypotheses on $\gamma \left(t\right)$, $\lambda \left(t\right)$, $q\left(t\right)$ we prove some stability results which relate the asymptotic behavior of the solutions of ${u}^{\text{'}\text{'}}+\gamma \left(t\right){u}^{\text{'}}+(q\left(t\right)+\lambda \left(t\right))u=0$ to the asymptotic behavior of the solutions of ${u}^{\text{'}\text{'}}+q\left(t\right)u=0$.

The paper is devoted to the question whether some kind of additional information makes it possible to determine the fundamental matrix of variational equations in ${\mathbb{R}}^{3}$. An application concerning computation of a derivative of a scalar Poincaré mapping is given.

In this paper we deal with a model describing the evolution in time of the density of a neural population in a state space, where the state is given by Izhikevich’s two - dimensional single neuron model. The main goal is to mathematically describe the occurrence of a significant phenomenon observed in neurons populations, the synchronization. To this end, we are making the transition to phase density population, and use Malkin theorem to calculate...

The behavior of the approximate solutions of two-dimensional nonlinear differential systems with variable coefficients is considered. Using a property of the approximate solution, so called conditional Ulam stability of a generalized logistic equation, the behavior of the approximate solution of the system is investigated. The obtained result explicitly presents the error between the limit cycle and its approximation. Some examples are presented with numerical simulations.