From spectral to nonlinear instability

An important property of a steady state of a dynamical system is its stability. Let x(t) be the state of the system at time t and let x_0 be a steady state. For a system of ODE these are points in Euclidean space while for more general dynamical systems they are functions which can be thought of as points in suitably chosen function spaces. In general it may be useful to have more than one function space in mind when considering a given dynamical system. First I will concentrate on the ODE case. It is possible to linearize the system about x_0 to get a linear system \dot y=Ay. The steady state x_0 is said to be linearly stable when the origin is stable for the linearized system. Since the linear system is simpler than the nonlinear one we would ideally like to be able to use linear stability as a criterion for nonlinear stability. In general the relation between linear and nonlinear stability is subtle even for ODE. We can go a step further by trying to replace linear stability by spectral stability. There are relations between eigenvalues of A with positive real parts and unstable solutions of the linearized system. Again there are subtleties. Nevertheless there are two simple results about the relation between spectral stability and nonlinear stability which can be proved for ODE. The first is that if there is any eigenvalue of A with positive real part then x_0 is nonlinearly unstable. The second is that if all eigenvalues of A have negative real parts then x_0 is nonlinearly stable, in fact asymptotically stable. These two results are far from covering all situations of interest but at least they do define a comfortable region which is often enough. In what follows I will only consider the first of these two results, the one asserting instability.

Up to this point I have avoided giving precise definitions. So what does nonlinear instability of x_0 mean? It means that there is a neighbourhood U of x_0 such that for each neighbourhood W of x_0 there is a solution satisfying x(0)\in W and x(t)\notin U for some t>0. In other words, there are solutions which start arbitrarily close to x_0 and do not stay in U. How can this be proved? One way of doing so is to use a suitable monotone function V defined on a neighbourhood of x_0. This function should be continuously differentiable and satisfy the conditions that V(x_0)=0, V(x)>0 for x\ne x_0 and \dot V>0 for x\ne x_0. Here \dot V is the rate of change of V along the solution. Let \epsilon be sufficiently small so that the closed ball \overline{B_\epsilon (x_0)} is contained in the domain of definition of V. We will take this ball to be the neighbourood U in the definition of instability. Let M be the maximum of V on \overline{B_\epsilon (x_0)}. Thus in order to show that a solution leaves U it is enough to show that V exceeds M. Consider any solution which starts at a point of V other than x_0 for t=0. The set where V(x)<V(x_0) is open and the solution can never enter it for t>0. The intersection of its complement with U is compact. Thus \dot V has a positive minimum there. As long as the solution does not leave U we have \dot V(x(t))\ge m. Hence V(t)\ge V(0)+mt. This implies that if the solution remains in U for all t>0 then V(x(t)) eventually exceeds M, a contradiction. This result can be generalized as follows. Let Z be an open set such that x_0 is contained in its closure. Suppose that we have a function V which vanishes on the part of the boundary of Z intersecting U and for which \dot V>0 on Z except at x_0. Then x_0 is nonlinearly unstable with a proof similar to that just given.

Now it will be shown that if A has an eigenvalue with positive real part a function V with the desired properties exists. We can choose coordinates so that the steady state is at the origin and that the stable, centre and unstable subspaces at the origin are coordinate subspaces. The solution can be written in the form (x,y,z) where these three variables are the projections on the three subspaces. Then A is a direct sum of matrices A_+, A_{\rm c} and A_-, whose eigenvalues have real parts which are positive, zero and negative respectively. It can be arranged by a choice of basis in the centre subspace that the symmetric part of A_c is as small as desired. It can also be shown that because of the eigenvalue properties of A_+ there exists a positive definite matrix B_+ such that A_+^TB_++B_+A_+=I. For the same reason there exists a positive definite matrix B_- such that A_-^TB_-+B_-A_-=-I. Let V=x^TB_+x-y^Ty-z^TB_-z. Then \dot V=x^Tx+z^Tz-y^T(A_c^T+A)y+o(x^Tx+y^Ty+z^Tz). The set U is defined by the condition V>0. There y^Ty\le Cx^Tx for a positive constant C. On this region \dot V\ge\frac12(x^Tx+z^Tz)+o(|x|^2+|z|^2), where we profit from the special basis of the centre subspace mentioned earlier. The quadratic term in y which does not have a good sign has been absorbed in the quadratic term in x which does. This completes the proof of nonlinear instability. As they stand these arguments do not apply to the infinite-dimensional case since compactness has been used freely. A discussion of the infinite-dimensional case will be postponed to a later post.

One Response to “From spectral to nonlinear instability”

  1. A model for the Calvin cycle with diffusion | Hydrobates Says:

    […] where ODE are coupled to a diffusion equation. This can be proved using a method discussed in a previous post which allows nonlinear instability to be concluded from spectral instability. We prove the spectral […]

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