## The Routh-Hurwitz criterion

I have been aware of the Routh-Hurwitz criterion for stability for a long time and I have applied it in three dimensions in my research and tried to apply it in four. Unfortunately I never felt that I really understood it completely. Here I want to finally clear this up. A source which I found more helpful than other things I have seen is https://www.math24.net/routh-hurwitz-criterion/. One problem I have had is that the Hurwitz matrices, which play a central role in this business, are often written in a form with lots of … and I was never sure that I completely understood the definition. I prefer to have a definite algorithm for constructing these matrices. The background is that we would like to understand the stability of steady states of a system of ODE. Suppose we have a system $\dot x=f(x)$ and a steady state $x_0$, i.e. a solution of $f(x_0)=0$. It is well-known that this steady state is asymptotically stable if all eigenvalues $\lambda$ of the linearization $A=Df(x_0)$ have negative real parts. This property of the eigenvalues is of course a property of the roots of the characteristic equation $\det(A-\lambda I)=a_0\lambda^n+\ldots+a_{n-1}\lambda+a_n=0$. It is always the case here that $a_0=1$ but I prefer to deal with a general polynomial with real coefficients $a_i, 0\le i\le n$ and a criterion for the situation where all its roots have negative real parts. It is tempting to number the coefficients in the opposite direction, so that, for instance, $a_n$ becomes $a_0$ but I will stick to this convention. Note that it is permissible to replace $a_k$ by $a_{n-k}$ in any criterion of this type since if we multiply the polynomial by $\lambda^{-n}$ we get a polynomial in $\lambda^{-1}$ where the order of the coefficients has been reversed. Moreover, if the real part of $\lambda$ is non-zero then it has the same sign as the real part of $\lambda^{-1}$. I find it important to point this out since different authors use different conventions for this. It is convenient to formally extend the definition of the $a_i$ to the integers so that these coefficients are zero for $i<0$ and $i>n$.

For a fixed value of $n$ the Hurwitz matrix is an $n$ by $n$ matrix defined as follows. The $j$th diagonal element is $a_j$, with $1\le j\le n$. Starting from a diagonal element and proceeding to the left along a row the index increases by one in each step. Similarly, proceeding to the right along a row the index decreases by one. In the ranges where the index is negative or greater than $n$ the element $a_n$ can be replaced by zero. The leading principal minors of the Hurwitz matrix, in other words the determinants of the submatrices which are the upper left hand corner of the original matrix, are the Hurwitz determinants $\Delta_k$. The Hurwitz criterion says that the real parts of all roots of the polynomial are negative if and only if $a_0>0$ and $\Delta_k>0$ for all $1\le k\le n$. Note that a necessary condition for all roots to have negative real parts is that all $a_i$ are positive. Now $\Delta_n=a_n\Delta_{n-1}$ and so the last condition can be replaced by $a_n>0$. Note that the form of the $\Delta_k$ does not depend on $n$. For $n=2$ we get the conditions $a_0>0$, $a_1>0$ and $a_2>0$. For $n=3$ we get the conditions $a_0>0$, $a_1>0$, $a_1a_2-a_0a_3>0$ and $a_3>0$. Note that the third condition is invariant under the replacement of $a_j$ by $a_{n-j}$. When $a_0a_3-a_1a_2>0$, $a_0>0$ and $a_3>0$ then the conditions $a_1>0$ and $a_2>0$ are equivalent to each other. In this way the invariance under reversal of the order of the coefficients becomes manifest. For $n=4$ we get the conditions $a_0>0$, $a_1>0$, $a_1a_2-a_0a_3>0$, $a_1a_2a_3-a_1^2a_4-a_0a_3^2>0$ and $a_4>0$.

Next we look at the issue of loss of stability. If $H$ is the region in matrix space where the Routh-Hurwitz criteria are satisfied, what happens on the boundary of $H$? One possibility is that at least one eigenvalue becomes zero. This is equivalent to the condition $a_n=0$. Let us look at the situation where the boundary is approached while $a_n$ remains positive, in other words the determinant of the matrix remains non-zero. Now $a_0=1$ and so one of the quantities $\Delta_k$ with $1\le k\le n-1$ must become zero. In terms of eigenvalues what happens is that a number of complex conjugate pairs reach the imaginary axis away from zero. The generic case is where it is just one pair. An interesting question is whether and how this kind of event can be detected using the $\Delta_k$ alone. The condition for exactly one pair of roots to reach the imaginary axis is that $\Delta_{n-1}=0$ while the $\Delta_k$ remain positive for $k. In a paper of Liu (J. Math. Anal. Appl. 182, 250) it is shown that the condition for a Hopf bifurcation that the derivative of the real part of the eigenvalues with respect to a parameter is non-zero is equivalent to the condition that the derivative of $\Delta_{n-1}$ with respect to the parameter is non-zero. In a paper with Juliette Hell (Math. Biosci. 282, 162), not knowing the paper of Liu, we proved a result of this kind in the case $n=3$.