Total Poisson boundedness of solutions of $\mathcal{P}$-perturbed complex systems of differential equations

We introduce the concept of a $\mathcal{P}$-perturbed system and, in particular, $\mathcal{P}$-perturbed complex system. Based on the method Lyapunov functions we obtain the sufficient condition of total Poisson boundedness of solutions of $\mathcal{P}$-perturbed systems with respect to any linear system with constant coefficients. Based on the method of vector Lyapunov functions and the above condition we obtain sufficient conditions of total Poisson boundedness of solutions of $\mathcal{P}$-perturbed complex system and solutions of $\mathcal{P}$-perturbed complex systems with feedback loop.