# Perron-Frobenius theorem applied to continuous-time dynamical systems

This is a question about terminology rather than mathematics per se.

I'm publishing a series of papers in which I make use of a fairly basic result that allows me to apply the Perron-Frobenius theorem in a case where the matrix is not non-negative but has negative entries only on its diagonal. I always end up having to explain the result in a footnote, and I'm wondering whether there's a standard name for it so that I can refer readers elsewhere instead.

The matrix arises from making a linear approximation to a dynamical system in which none of the variables can become negative. We end up with $\mathbf {\dot x} = J\mathbf x$, where $J$'s off-diagonal elements are all non-negative, but the diagonal elements may be negative.

If we solve this for a small finite $\delta t$ we get $\mathbf x_{t+\delta t} = e^{J\delta t}\mathbf x_t$. If $\delta t$ is small then $e^{J\delta t} \approx 1 + J\delta t$, whose entries are all non-negative as long as $\delta t$ is small enough. Thus we can apply thw Perron-Frobenius theorem to this class of continuous dynamical systems in the same way that it's often applied to discrete dynamical systems.

Moreover, the eigenvectors of $J$ are the same as the eigenvectors of $e^{J\delta t}$ and its eigenvalues are proportional to the logarithms of the eigenvalues of of $e^{J\delta t}$, so we can easily speak in terms of the eigenvalues and eigenvectors of $J$ itself. (With the main difference being that we're now talking about the eigenvalue with largest real part rather than the largest absolute value.)

This all seems like it should be fairly standard, but I haven't come across it anywhere. Is there a standard resource that explains it, or should I just keep explaining it from scratch every time I summarise my results that use it?

Note that I'm not asking for extra formal details beyond what I've described above - this is a tiny step in an applied paper, aimed at an audience who are not primarily mathematicians (even applied ones) and may not know much beyond basic linear algebra. In talks I've found that the audience tends to understand the application of Perron-Frobenius to discrete dynamical systems, and they understand the idea of discretising a continuous dynamical system, but putting the two together like this always requires a bit of explanation, and I'm asking where I can send people to read up on it.

• When you are talking about applying Perron-Frobenius theory to a matrix what kind of result you want to get? – demitau Mar 15 '15 at 22:50
• @demitau mostly the usual one, that for an irreducible matrix there is a unique leading eigenvalue (in this case the one with greatest real part), which is real, whose corresponding eigenvector has only positive components. For dynamical systems of this form this implies there is a unique vector with positive entries to which all trajectories converge, up to a multiplicative factor, and if the leading eigenvalue is positive everything increases exponentially. Then there's all the usual stuff about how you can block-diagonalise reducible matrices and therefore use this to say things about them. – Nathaniel Mar 16 '15 at 7:49
• I thought usually pepople study not exactly matrix $J$ but its resolvent instead. Then you can apply standard techniques. – demitau Mar 17 '15 at 12:26