Asymptotic behavior of solutions of systems of dynamic equations on time scales in a set whose boundary is a combination of strict egress and strict ingress points

In this paper we study the asymptotic behavior of solutions of nonlinear dynamic systems on time scales of the form $y^{\mathrm{\Delta }}(t)=f(t\text{,}y(t))\text{,}$ where $f:‐{T}\times {‐{R}}^n\to {‐{R}}^n$ and $‐{T}$ is a time scale. For a given set $\mathrm{\Omega }\subset ‐{T}\times {‐{R}}^n$ , we formulate the conditions for function f, which guarantee that at least one solution y of the above system stays in $\mathrm{\Omega }$ . The dimension of the space of initial data generating such solutions is discussed and perturbed linear systems are considered as well. A linear system with singularity at infinity is considered as an example.