Topological conditions for the existence of bounded solutions of quasihomogeneous problems

We consider a system of differential equations $\dot x=P(x,t)+X(x,t)$, $(x,t)\in R^n\times R$ where $P\in C^1(R^n\times R)$ and is a positively homogeneous function of $x$ of degree $m$, larger than one, and the function $X$ is small in comparison with $P$ at infinity. In terms of the Lyapunov–Krasovskii function of the corresponding homogeneous system a certain submanifold of the unit sphere is defined. It is shown that if this submanifold is not contractible, then the quasihomogeneous system being considered has at least one bounded solution. The proof is based on the topological principle of Wazewski.