When is a dynamical system empty of information?

This post is concerned with dynamical systems in the sense of systems of ordinary differential equations. It is well-known that as soon as the dimension of the system is at least three general dynamical systems can include very complicated behaviour (chaos, strange attractors etc.) which defy detailed understanding in the absence of extra restrictions. In many applications of dynamical systems to the real world (for instance to biology) we encounter the following situation. A dynamical system is postulated which involves a lot of unknowns (in particular more than three) and depends on many parameters. There is only limited information about the parameters which means that in reality we are confronted with a large class of systems. Moreover there is only a limited amount of information about which initial data lead to solutions relevant to the application of interest. This means that in effect a large class of solutions is involved when an attempt is made to study the problem. The door is wide open to the daunting complexity of general solutions of general systems. The standard reaction is to make some choice of parameters and initial data and solve for the evolution on the computer. Since only a finite number of cases can be computed there is no guarantee that what comes out is typical of what happens in general or relevant for the application.

It might be hoped that general properties of dynamical systems which arise in particular wide classes of applications might reduce some of these difficulties. An example of a class of this type is defined by the systems which describe the competition of species in ecology. In 1976 Stephen Smale (J. Math. Biol. 3, 5) showed in a three-page paper that the restriction to this type of system does not, on its own, lead to any improvement whatsoever. In more detail what he did was the following. He considered a system of ODE of the form \dot x_i=x_if_i(x) where the f_i are smooth and \frac{\partial f_i}{\partial x_j}<0 for all i,j. The inequality expresses the competitive property while the factor x_i on the right hand side ensures that if the population of a species is initially zero it stays zero. The region where all x_i are non-negative, which is the biologically interesting region, is invariant. Finally he assumes that the functions f_i are negative outside a large ball so that solutions cannot escape to infinity. Let \Delta_1 be the set where all x_i are non-negative and \sum_i x_i=1. Smale shows that there is a dynamical system in the class under discussion here where \Delta_1 is an invariant subset, all solutions converge to \Delta_1 as t\to\infty and the dynamical system on \Delta_1 is arbitrary. Thus any complication of a general dynamical system of dimension n can be built into a model of the dynamics of n+1 competing species. There is no fundamental simplification if n is at least three. Soon afterwards an analogous result was proved for a class of systems in chemical physics describing continuous flow stirred tank reactors by Perelson and Wallwork (J. Chem. Phys. 66, 4390). A paper of Kaplan and Yorke (American Naturalist 111, 1030) mentions the relevance of Smale’s result to competitive exclusion, a topic mensioned in a previous post.

It is not my intention here to spread pessimism. I hope and believe that dynamical systems can be of immense value in many scientific applications and in biology and medicine in particular. I just want to emphasize that the act of writing down a dynamical system satisfying a few general assumptions is no guarantee that some interesting information about a scientific problem has been implemented. I would add that being able to calculate a few solutions on the computer is not a sufficient reason to be confident that something has been achieved. It should be a matter of priority to develop ways of ensuring that systems proposed do contain information.

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5 Responses to “When is a dynamical system empty of information?”

  1. hydrobates Says:

    I discovered from the book ‘Monotone dynamical systems, an introduction to the theory of competitive and cooperative systems’ by H. Smith that there is a nice result for systems which are cooperative. These differ from competitive systems only by a sign. Reversing the sign of time converts one into the other and so it did not occur to me when I wrote the post that there could be a big difference. In the example of Smale the complicated dynamics occurs in an invariant manifold which describes the late-time behaviour of solutions. The early-time behaviour of cooperative systems is just as complicated. In contrast, the late-time behaviour of cooperative systems has better properties. In that case the \omega-limit set of a generic solution consists of stationary points.

  2. Martín Says:

    Hello. Your last sentence reads “It should be a matter of priority to develop ways of ensuring that systems proposed do contain information.”

    Along your blog entry, you never define or propose a definition for information content, a some sort of scale that measure how much information a systems of ODE’s is potentially capable to have.

    Is there such a “concept” for ODEs, i. e. a concept of information independent of how well the ODEs solutions “fit” the experimental data?

  3. hydrobates Says:

    In this post I was using the word ‘information’ in an informal, intuitive way. I did not intend to suggest any technical use of the word and I am not aware of a relevant concept of this kind.

  4. hydrobates Says:

    It is implicit in the above remarks about the difference between competitive and cooperative systems that the variables x_i are assumed positive. Otherwise systems could be transformed from one class to the other just by changing the sign of all the x_i.

  5. The Perelson-Wallwork theorem « Hydrobates Says:

    [...] this post I will explore some connections between two themes I have previously written about, dynamical systems empty of information and chemical reaction network theory. In the first of these posts I mentioned a paper of Perelson [...]

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