Routh-Hurwitz Criterion

Question

I am studying a dynamical system with 4 equations. When I evaluate the Jacobian Matrix in a critical point and I see that the trace is zero, how can I use the Routh-Hurwitz Criterion to obtain some conclusion about the stability of the critical point? Thanks.

Answer 1

The trace is the sum of the eigenvalues (counted by algebraic multiplicity). If the trace is $0$, at least one eigenvalue must have nonnegative real part. If some eigenvalue has positive real part, the critical point is unstable. However, it is possible that all eigenvalues have real part $0$, e.g. the matrix could be $$\pmatrix{0 & 1 & 0 & 0\cr -1 & 0 & 0 & 0\cr 0 & 0 & 0 & -1\cr 0 & 0 & 1 & 0\cr}$$ in which case the linearized system is stable but not asymptotically stable, but the nonlinear system could be either stable or unstable, depending on the nonlinear terms.

Answer 2

Assuming you are talking about continuous system, the trace being 0 just means that the flow is volume preserving. E.g. trace of any fixed point in a Hamiltonian systems is 0, but that fixed point could be a Saddle type (unstable) or center type (marginally stable, 0 real part) point, or a combination of 2 (in higher dimensions, e.g. a Saddle X center in 4 dimensions), and so-on.