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 where the are smooth and for all . The inequality expresses the competitive property while the factor on the right hand side ensures that if the population of a species is initially zero it stays zero. The region where all are non-negative, which is the biologically interesting region, is invariant. Finally he assumes that the functions are negative outside a large ball so that solutions cannot escape to infinity. Let be the set where all are non-negative and . Smale shows that there is a dynamical system in the class under discussion here where is an invariant subset, all solutions converge to as and the dynamical system on is arbitrary. Thus any complication of a general dynamical system of dimension can be built into a model of the dynamics of competing species. There is no fundamental simplification if 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.