Proving global stability of steady states of dynamical systems

In this post I want to discuss some techniques for proving that solutions of dynamical systems converge to steady states. One frequently used and powerful method of proving results of this type is to find a Lyapunov function. The problem with that is the lack of systematic ways of finding such functions even when they exist. Here I discuss a quite different approach. One way to try to prove that a solution converges to a steady state is to prove that it does not do anything else. Here I will restrict to solutions which are bounded (also away from the boundary of the domain of definition of the system, if there is one) so that they should converge to something. One obvious possibility is that the solution might converge to a periodic solution. Thus it is important to have criteria for ruling out periodic solutions. For systems of two equations there is a well-known criterion of this type due to Bendixson. A vector field on the plane whose divergence is everwhere positive (or everywhere negative) has no periodic solutions. The image of the solution is a Jordan curve and the integral of the divergence over its interior is positive (negative). On the other hand the divergence theorem shows that this is equal to a boundary integral and since the vector field is tangent to the boundary this integral is zero. Thus the existence of a periodic solution leads to a contradiction. This can be generalized in two ways. Instead of the whole plane the domain of the vector field could be a simply connected open subset. The vector field can be multiplied by a positive function, which does not affect the presence of periodic solutions. The resulting criterion is named after Dulac and the function which occurs is called a Dulac function. Other possibilities are homoclinic solutions and heteroclinic cycles. These define simple closed curves in the plane and can be ruled out by the argument just given in the presence of a Dulac function. Suppose now that all steady states are isolated. Then Poincaré-Bendixson theory shows that the possibilities we have considered are the only ones. To conclude, under the conditions we have considered on a dynamical system in two dimensions the existence of a Dulac function implies that every solution converges to a steady state.

The aim now is to find some generalization of this story to higher dimensions. The strategy consists of two parts. The first is a criterion for the absence of periodic solutions generalizing that of Bendixson. The second is to show that if an integral curve of a vector field returns arbitrarily close to its starting point there exists a small local perturbation of class $C^1$ of the vector field which admits a closed integral curve. This statement is known as Pugh’s closing lemma. If now the applicability of the Bendixson criterion is preserved under small perturbations this gives us a way of approaching the desired goal. From now on I concentrate on the Bendixson criterion. In the two-dimensional case the integral which was calculated can be thought of geometrically in the following way. Let $S$ be the interior of the periodic solution. Now apply the flow of the vector field. The integral is the rate of change of the area of $S$ under the flow. Since $S$ does not move under the flow this rate of change is zero. Consider now a vector field on an open subset $D$ of $n$-dimensional Euclidean space. A closed integral curve of this vector field can be thought of a mapping from the unit circle in the plane to $D$. If it is assumed that $D$ is simply connected then (ignoring questions of regularity for the moment) this mapping extends to a mapping of the unit disc to $D$ with image $S$. We now want to repeat what was done in the plane and consider how the area of $S$ changes with the flow. On the one hand we would like to do a computation which shows that the rate of change of the area is non-zero. On the other hand we would like to choose $S$ to be a surface of minimal area with the given boundary. These two elements together lead to a contradiction and rule out the existence of a periodic solution.

The rate of change of the area can be estimated using a suitable generalization of the divergence of the vector field. It is necessary to obtain information about the way in which two-dimensional area elements are affected by the flow. The necessary considerations can be found in a paper of Muldowney (Rocky Mountain Journal of Mathematics 20, 857). I cannot explain all details here and I will confine myself to mentioning some important ideas involved. We start with an autonomous system $\dot x=f(x)$ of ODE. Linearizing about a solution gives a non-autonomous linear system $\dot y=A(t)y$, the variational equation. It describes how line elements are affected by the flow. To see how area elements are affected we must take the wedge product of two linearized solutions. It satisfies an equation $\dot z=A^{[2]}(t)z$ where $A^{[2]}$ is called the second additive compound of $A$. There is a direct algebraic formula which determines $A^{[2]}$ from $A$. What is still needed is a way of estimating the solutions of these linear equations. This is done by using the Lozinskii logarithmic norm. I have discussed this in a previous post under the name ‘matrix measure’. Unfortunately I still do not have a good intuition for what it means. The definition is quite flexible due to the possibility of starting from different norms on the space of matrices. Concretely, one condition ensuring the absence of periodic solutions is that the Lozinskii norm of the second additive compound of $Df$ is negative. With a suitable choice of norm on the space of matrices this is equal to the sum of the two largest eigenvalues of the symmetric part of $Df$.