2.5 Comparing Mathematical Representations
2.5.2 Qualitative Behaviour and Selective Accuracy
Now that we have narrowed down what a good model is, we should try to find out when a model is ‘good enough.’ Implicitly, this problem requires that we determine when a model is better than another, or just as good as another, i.e., it involves the comparison of the accuracy of different model equations. As a part of the theory of dynamical systems, applied mathematicians have
developed many concepts that capture when two models are essentially similar. Some of them are rather simple, others are more complex and refined.
Models can be built with different objectives in mind. In some cases, it is important to obtain precise numerical information concerning the solution— i.e., the states of the system on a given time interval—but in others we are only concerned with properties of the solution that only indirectly depend on the actual values of the states at times.35 In other words, we are sometimes
not interested in knowing exactly what happens in a system, but rather only in what kinds of thing happen in a system. In this case, we will consider a model to be selectively accurate (and therefore a good representation) if it implies the right kinds of things. This is whyqualitative analyses based on the general theory of dynamical systems plays an important role in determining what a ‘good’ model is.
Let us begin our discussion of the qualitative aspect with a simple case. Suppose a model is characterized by the following model equation:
˙
x=x2−t, x0 =x(0) =−
1
2. (2.10)
This is a differential equation of the state x with respect to time, and it also contains an initial condition. Together, they determine what will happen in the system. What would happen if the initial condition were instead 0, or
−1, or −3? What if the same initial conditions were not given at t = 0 but rather at t = 3, t = 5 or t = 7? As it turns out, it would essentially change nothing, except for a very short initial time interval. The solution of equation (2.10) is displayed in figure 2.18 in bold, together with many other solutions using different initial values. We see that the trajectories all converge to the same one extremely rapidly. As a result, we can claim that the solution is insensitive to perturbations of the parameter x0. When this is the case, we
can say that two model equations with different values ofx0 are equally good,
unless the behaviour of interest concerns the early time interval. Accordingly, 35Examples of such properties of solution include being periodic, approaching a limit cycle, having vanishing terms, being bounded,etc. In general, it includes all properties related to the structure of attractors.
0 2 4 6 8 10 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0
Figure 2.18: Equation (2.10) is an extremely robust differential equation. we say that both the reference model and perturbed versions of the model are selectively accurate, both quantitatively and qualitatively.
In the above example, the sense in which the two model equations are equally good can be made more precise. Suppose we have a differential equa- tion ˙x = f(t, x) and two initial conditions x0 = α and x0 = β, and further
suppose that y(t) and z(t) are their respective solutions. If we have
lim
t→∞y(t) =x(t) and tlim→∞z(t) =x(t),
then we say that the two model equations have solutions that converge in the limit t → ∞. Often, the solution of differential equations can be written as a sum of two solutions as follows:
x=xtransient+xsteady state
The transient solution is the one that makes the trajectories differ early on, but then it vanishes as time increases and the steady state solution becomes the dominant one. Differential equations with the same steady state solution will thus be considered to be equally good, provided that the transient behaviour is not crucial. Ignoring transient behaviour is a typical case of selective accuracy.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −5
0 5 10
Figure 2.19: Qualitative change in the solution of ˙x=x2−t atx(0) = �3 1/2
(which is the dotted line).
In the example above, we would thus consider all systems for which x(0) < 3
�
1/2 equivalent in this sense. However, for x(0)≥ �3 1/2, the systems could
not be considered equivalent, since they would diverge away from each other, as shown in figure 2.19. At x(0) =�3 1/2, there is a qualitative change taking
place in the behaviour of the system, i.e., there is a bifurcation. Moreover, note that, if we set a value x(0) less than, but close to the critical value, a perturbation of the system could easily push the system on the other side of the bifurcation line; thus great care would need to be taken in this region. As a result, we would not say that both the reference model ˙x = x2 − t,
x(0) = �3 1/2
−�, and slightly perturbed models are selectively accurate. Above, we examined the sensitivity to changes in the initial conditions. What would happen if we made modifications in the functionf in a differential equation ˙x=f(t, x) instead? Consider the family of coupled two-dimensional linear systems of ordinary differential equations given by
˙
x=ax+by
˙
plus some initial conditions x0 and y0. It can be rewritten in matrix-vector notation as ˙x=Ax, where x= � x y � and A= � a b c d � .
When we change the values of the entries of the matrixA, what will happen to the behaviour of the system? One way to address this question is by looking at the type of attractor that obtains in the limit when t → ∞. As it turns out, it entirely depends on the eigenvalues of the matrix A, which are given by
λ= 1 2
�
Tr A−�Tr2A−4 detA�,
where TrA is the trace of the matrixAand detA is its determinant.36 Thus,
the structure of the attractor to which the system tends as t → ∞ is fully determined by the invariant quantities Tr A and detA. The whole situation can be summarized in a bifurcation diagram, as in figure 2.20. There, we see that there are six distinct possibilities for the structure of the attractor (sink, source, saddle, cycle, spiral sink, and spiral source). For instance, if Tr A > 0, detA > 0, but Tr2A > 4 detA, we have the rightmost case, namely, a source. Since the computation of those two invariant quantities is straightforward, it is not problematic to find out whether changes in a, b, c
and d changes the qualitative behaviour of the system. To the extent that the limit behaviour of the system is of interest, it is thus justified to consider two systems in the same region of the bifurcation diagram to be equally good representations of a system. In fact, when this is the case, mathematicians call the system topologically equivalent, or topologically conjugate. Specifically, this notion of equivalence is based on continuously varying aspects of the models; two models are topologically conjugate if there is a homeomorphism 36The reason for which the eigenvalues of A are key is this. Using the eigenvalue de- composition A = QΛQ−1, where Q is an orthogonal matrix of eigenvectors and Λ is a diagonal matrix of eigenvalues, we can write ˙x=Axas ˙x=QΛQ−1x. Making the substi- tutiony=Q−1xresults in thedecoupled system of equations ˙y=Λy, whose solutions are
y(t) =y0eλt. Moreover, note that since those are straight-line solutions, either the largest λwill dominate the behaviour asymptotically, or there will be periodic solutions.
Tr
det
Figure 2.20: An example of bifurcation analysis that completely characterizes the qualitative behaviour of all possible two-dimensional linear systems of or- dinary differential equations in terms of two invariant quantities (the trace and the determinant) only.
relating their solutions. There is a long tradition initiated by Poincar´e (1892) and Birkhoff (1966) that uses the general theory of dynamical systems to qualitatively characterize the equivalence of systems in this way. It amounts to defining equivalence of systems in a parameter space, rather than by direct reference to the states of the system over a given time interval. Here again, the relevant notion of comparative ‘goodness’ of models should be understood as selective accuracy.
Many systems turn out to have some features that are insensitive to per- turbations, while other are. Thus, even if the model is such that if there is any distortion, the information extracted about the states will be inaccurate, those distortion can be introduced to increase the manageability and tractability of the model concerning other properties. For instance, in the case of a chaotic system such as the Lorenz system (see figure 2.21), perturbation methods can- not accurately tell us what the states are, but they can very reliably tell us what is the dimension of the chaotic attractor, what its shape is, and the like.
0 10 20 30 40 50 −20 −10 0 10 20 −30 −20 −10 0 10 20 30
Figure 2.21: Phase portrait of the Lorenz system.
Those are properties insensitive to perturbations. As a result, it is crucial to bear in mind the questions we wish to answer and the behaviour of interest that must be understood in order to assess representations.