Routh-Hurwitz Criterion in the Examination of Eigenvalues of a System of Nonlinear Ordinary Differential Equations

In stability analysis of nonlinear systems, the character of the eigenvalues of the Jacobian matrix (i. e. , whether the real part is positive, negative, or zero) is needed, while the actual Value of the eigenvalue is not required. We present a simple algebraic procedure, based on the Routh-Hurwitz criterion, for determining the character of the eigenvalues without the need for evaluating the eigenvalues explicitly. This procedure is illustrated for a system of nonlinear ordinary differential equations we have studied previously. This procedure is simple enough to be used in computer code, and, more importantly, makes the analysis possible even for those cases where the secular equation cannot be solved.


I. INTRODUCTION
When studying systems of nonlinear ordinary differential equations, it is often useful to examine the eigenvalues of the associated linearized system. For example, consider the system of nonlinear ordinary differential equations given by W=H W+G(W), where W= W(t) is a column n vector, H is an n &&n matrix, and G(W) is a nonlinear vector-valued function. G(W) is defined such that G(W)/~~W~~is continuous and vanishes for W~zero. The stability of solutions to a system (I) satisfying these conditions can be determined using the eigenvalues of H (I) The equilibrium solution W(t) =0 is asymptotically stable if all the eigenvalues of H have negative real part.
(2) The equilibrium solution W(t):-0 is unstable if at least one eigenvalue of H has positive rea1 part.
(3) The stability of the equilibrium solution W(t)=-0 cannot be determined from H alone if all the eigenvalues of H have a real part less than or equal to 0 but at least one eigenvalue of H has a zero real part.
Thus, we do not really need to know the numerical values of the eigenvalues: All we need to know is the character of each eigenvalue. (By character we mean whether the real part of the eigenvalue is positive, negative, or zero. } We note also that in practice it may be difficult, or impossible, to calculate the actual values of the eigenvalues. Finding the values of the eigenvalues is equivalent to finding the zeros of an nth-degree polynomial. For n &5 there exists no general procedure for determining the zeros of an nth-degree polynomial (and, equivalently, the values of the eigenvalues), except for special cases.
We outline here a simple algebraic procedure for quick- where A, B, and C are functions of time t. The functions A, B, and C are proportional to the amplitudes of the terms of the truncated Fourier series. In (2), d) ---I /R, dp -- where k and I are constants related to the wave numbers of terms of the Fourier series, m is a constant related to the assumed boundary-layer flow along the plate, and R is a variable. R is the Reynolds number of the fluid. In our previous work we used k =0.62, I =0.07, and m = -0.064, and examined the behavior of system (2) as we varied R in the range 1 to 40. ly determining the character of the eigenvalues. The procedure is based on the Routh-Hurwitz criterion for determining the character of zeros of an nth-degree polynomial.
We illustrate the procedure by applying it to a system of nonlinear ordinary differential equations that we have Step A. We change the system (2) so that it resembles system (1). Specifically, we want S'=-0 to be the equilibrium solution; also we want to split the system into linear and nonlinear parts corresponding to the matrix H and the function G.
(1) Identify the equilibrium points of (2), that is, those points where A, B, and C are zero. In system (2) there are five equilibrium points, which we represent as P;=(A;,B;,C;) for i=0, 1,2, 3,4. These five equilibrium points are for our values of k, l, and m.
This results in a new nonlinear system: equilibrium points are real, but P1=P3 and P2 =P4. For R greater than R, all the equilibrium points are real and distinct. The value of R, can easily be computed by setting the radicand in A1 or A3 equal to zero; thus Notice that for R less than a critical value R"only Pp is real and P1 4 are all complex. For R equal to R, all the I as desired. Since system (5) is in the form of system (1), we could now determine the stability of the equilibrium solutions by examining the eigenvalues of the matrix H.
Step B. Start determining the eigenvalues of the matrix H. We do this in the usual way, by setting det(H -AI) =0. This yields the characteristic equation Using the definitions of di, dz, e, , ez, e3, f"fq, and the possible choices for A;, B;, and C; in Eq. (7) yields 0=~+ ""+"'~+ R ml(1k )A; (k +1) (k -1 )A; (4k'A, +2ml) . R (k'+ I') (8) This is the characteristic equation one would ordinarily solve to determine the eigenvalues of H. Equation (8) is a cubic equation in A. . A general solution to the cubic equation exists, but is very complicated. To obtain a general solution to (8), we would take the coefficients of the A, terms and substitute them into the general solution to the cubic. The result is an extremely complicated equation which obscures, rather than elucidates, the character of the eigenvalues as the value of R changes. Recall that we only desire to know the character of the eigenvalues. We do not need to solve (8) for 2, .
Step C. Use the Routh-Hurwitz criterion to determine the character of the solutions to (8). We first write our nth-degree polynomial in the form 0=apkn+a1gn -1+ ' ' +a. k"~+ +a" Comparing this form to equation (8), we can identify (4k A;+2ml) . R (k'+ l') (9) are no changes in sign of the sequence and no positive real parts of the eigenvalues; and (2) if the fourth term in the sequence is negative, there will be one sign change in the sequence and one positive real part of the eigenvalue.
Examining T3/T2 in (11) we see that the factor outside the parentheses is always positive since A; is always positive when A; is real. The factor inside the parentheses can be positive, negative, or zero. We find that this factor is only zero when Next we form the numbers Tp, T&, T2, T3 as

R(k +l )
Once we have constructed this sequence we can determine the character of the eigenvalues in this way (Routh-Hurwitz criterion): The number of roots with positive real parts of a real algebraic equation is equal to the number of sign changes in the sequence (11) above. [Our Eq. In (11), with real k and 1 and positive R, it is easy to see that the first three terms of the sequence are all positive. Therefore, the Routh-Hurwitz criterion tells us that (1) if the fourth term in the sequence is also positive, there which is identical to R, in (3). Stability of the equilibria can therefore only change at R =R, . Thus we have the following cases: Case I: R=R, . Then PI --P3 and P2 --P4 and Q3 is zero, implying that one eigenvalue [root of (8)] is zero, implying the other two eigenvalues [roots of (8)

III. CONCLUSIONS
We have seen that we can easily find the character of the eigenvalues of the matrix of the linearized system associated with a system of nonlinear ordinary differential equations. There is a step-by-step procedure which uses the Routh-Hurwitz criterion in determining the character of eigenvalues. This procedure would prove useful in any study which involves examining the eigenvalues of a system, especially a nonlinear system. The procedure is simple enough that it could be adapted to computer codes, without great loss of performance.
We wish to thank Professor Gerasimos Ladas for a suggestion which led to this study.