## 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) 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.