In this paper, we define \(\Psi \)-boundedness on time scales and we present necessary and sufficient conditions for the existence of at least one \(\Psi\)-bounded solution for the linear non-homogeneous matrix system \(x^{\Delta}=A(t)x + f(t)\), where f(t) is a \(\Psi\)-bounded matrix valued function on \({T}\) assuming that \(f\) is a Lebesgue \(\Psi\)-delta integrable function on time scale \({T}\). Finally we give a result in connection with the asymptotic behavior of the \(\Psi\)-bounded solutions of this system.

Assume that \((W, g_{1,\infty})\) is a nonautonomous discrete dynamical system given by sequences \((g_{m})_{m=1}^{\infty}\) of continuous maps on the space \((W,d)\). In this paper, it is proven that if \(g_{1, \infty}\) is topologically weakly mixing and satisfies that \(g_{1}^{n}\circ g_{1}^{m}=g_{1}^{n+m}\) for any \(n,m\in\{0,1,\ldots\}\), then it is distributional chaos in a sequence. This result extends the existing one.