Extension of the Perron-Frobenius theorem to homogeneous systems

This paper deals with a particular class of positive systems. The state components of a positive system are positive or zero for all positive times. These systems are often encountered in applied areas such as chemical engineering or biology. It is shown that for this particular class the first orthant contains an invariant ray in its interior. An invariant ray generalizes the concept of an eigenvector of linear systems to nonlinear homogeneous systems. Then sufficient conditions for uniqueness of this ray are given. The main result states that the vector field on an invariant ray determines the stability properties of the zero solution with respect to initial conditions in the first orthant. The asymptotic behavior of the solutions is examined. Finally, we compare our results to the Perron-Frobenius theorem, which gives a detailed picture of the dynamical behavior of positive linear systems.