The Genesis of Dynamic Systems Governed by Metzler Matrices.
HARVARD UNIV CAMBRIDGE MASS
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This paper studies the behavior in the neighborhood of the starting point of dynamic systems solutions of difference or differential equations whose Jacobians are Metzler matrices. A Metzler matrix is one whose off-diagonal elements are non-negative. A new concept, that of first-positivity of a sequence, is introduced a sequence is first-positive if its first non-zero element is positive. First-positivity holds for a sequence of matrices or vectors if it holds for each component. It is shown that the sequence of powers of a Metzler matrix is first-positive also, for each position in the matrix defined by row and column, the number of steps to the first non-zero entry in the sequence is equal to the minimum length of a chain from the row to the column through non-zero entries in the Metzler matrix. From this, it is possible to express a the time of first-positivity of a specific component of the solution to a system of difference equations and b the order of increase of a specific component of the solution to a system of differential equations in terms of the connectivity properties of the governing matrix, the specification of the positive components of the starting values, and the first-positivity properties of the forcing functions. Author
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