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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.

This paper is concerned with the oscillatory behavior of the damped half-linear oscillator ${\left(a\left(t\right){\phi }_p\left(x^{\text{'}}\right)\right)}^{\text{'}}+b\left(t\right){\phi }_p\left(x^{\text{'}}\right)+c\left(t\right){\phi }_p\left(x\right)=0$, where ${\phi }_p\left(x\right)={\left|x\right|}^{p-1}\mathrm{sgn}x$ for $x\in ℝ$ and $p>1$. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are...

This paper is concerned with the global asymptotic stability of the zero solution of the half-linear differential system $$x^{\text{'}}=-e\left(t\right)x+f\left(t\right){\phi }_{p^{*}}\left(y\right),y^{\text{'}}=-g\left(t\right){\phi }_p\left(x\right)-h\left(t\right)y,$$ where $p>1$, $p^{*}>1$ ($1/p+1/p^{*}=1$), and ${\phi }_q\left(z\right)={\left|z\right|}^{q-2}z$ for $q=p$ or $q=p^{*}$. The coefficients are not assumed to be positive. This system includes the linear differential system ${𝐱}^{\text{'}}=A\left(t\right)𝐱$ with $A\left(t\right)$ being a $2\times 2$ matrix as a special case. Our results are new even in the linear case ($p=p^{*}=2$). Our results also answer the question whether the zero solution of the linear system is asymptotically stable even when Coppel’s condition does not hold...