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 jth diagonal element is a_j. 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<n-1. 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.

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