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This paper is focused on the the behavior near the extinction time ofsolutions of systems of ordinary differential equations with a sublineardissipation term. Suppose the dissipation tern is a product of a linearoperator $A$ and a positively homogeneous scalar function $H$ of a negativedegree $-\alpha$. Then any solution with an extinction time $T_*$ behaves like$(T_*-t)^{1/\alpha}\xi_*$ as time $t\to T_*^-$, where $\xi_*$ is an eigenvectorof $A$. The proof first establishes the asymptotic behaviors of the "Dirichlet"quotient and the normalized solution. They are then combined with aperturbation technique that requires the function $H$ to satisfy some pointwiseH\"older-like condition. The result allows the higher order terms to be generaland the nonlinear function $H$ to take very complicated forms.

Behavior near the extinction time for systems of differential equations with sublinear dissipation terms