Handling Complex Dynamics

In the previous subsection, we considered the methods used to reduce the complexity of the structure of the real physical system, resulting in significant reduction in the structure of the model we need to deal with in order to study the dominant behaviour. There is another kind of complexity that must be handled properly for effective mod- eling, viz., complex dynamics. Both of the models constructed above (for the double pendulum and orbit modeling), and the physical systems modeled, exhibit dynamical chaos (extreme sensitivity to initial conditions).20 _{Chaos is not an all or nothing}

phenomenon, however. Some regions of the phase space of a dynamical system can be chaotic while others are not, and certain evolutionary phenomena can be studied in chaotic regions provided that the time scale of the process is shorter than the time scale where exponential divergence becomes significant, and even longer if accurate individual trajectories are not required, as we will see below. Both the double pen- dulum and the motion of objects in the solar system are examples of the former case, which is actually a property of most chaotic dynamical systems (Wisdom,1987). The

20_{More technically, the typical standard condition for dynamical chaos is generic, bounded expo-}

issue of the time scale over which chaotic effects become significant is a particular concern in the study of the orbits of NEO. Accordingly it is important to be able to determine the time scale over which chaotic behaviour becomes evident or significant. The double pendulum, however, easily exhibits significant chaotic behaviour, which requires a different approach, viz., the asking of different questions.

The problem of estimating the impact probability of a NEO requires that we are able to accurately predict the orbit of a NEO for a significant period into the future. This problem is not solved merely by ensuring that the data collected from the object has certain qualities, since the orbit of the object itself must be stable. This is an issue in the solar system because the orbits of objects are, in general, chaotic. Chaotic behaviour is actually a ubiquitous feature of hamiltonian dynamical systems, so it is a quite general concern for dynamical phenomena (Wisdom, 1987). Consequently, the ability to compute the (first) Lyapunov exponent λ, which characterizes the un- predictability for a dynamical system, for an object is essential in order to be able to know the characteristic time over which chaotic effects will become significant. The Lyapunov exponent measures the maximal log-rate of separation of infinitesimally close trajectories. Whenλ>0, nearby trajectories (on average) diverge exponentially and the dynamics are chaotic. The size of λ, then, determines the rate. The time

τe taken for the separation of two trajectories to increase by a factor of e is just the reciprocal of the Lyapunov exponent:

τe= 1

λ.

This is called the e-folding time, and is one measure of the characteristic time scale over which chaotic effects will begin to become significant. A measure of the char- acteristic time scale of chaos is called a Lyapunov time. The e-folding time is one definition of this among others, but the typical one in celestial mechanics. Conse- quently, we will refer to the e-folding time asthe Lyapunov time.

One of the significant features of chaotic motion for feasible epistemology is that it implies that precise trajectories can be tracked theoretically only locally to a certain period of time. Since chaos is a general property of hamiltonian dynamical systems, this is not merely a special consideration for some cases. Thus, the generic case

should be that knowledge is local to a limited range of conditions. The logical ap- proaches do not provide a conceptual language for such local knowledge. This could be seen as a result of the theory T syndrome, given the fact that even the classical existence and uniqueness theorem for nonlinear dynamical systems only apply locally. Local knowledge is a general issue with nonlinear problems, not even just for chaos. Consequently, an adequate conceptual framework for describing our knowledge of the world should deal generally with local knowledge, which is precisely what the local constraint logic we are using in this study does.

There is also an issue here for the view that model construction is deductive. For real solar system bodies, particularly NEO, analytic computation of the Lyapunov time is not feasible. Consequently, numerical methods must be used to compute this time. This shows that a key part of feasible inference for nonlinear phenomena is to be able to estimate how long the results of a model will be accurate. This does not emerge as a significant concern on the semantic views considered in the last chapter. A common numerical technique used in NEO modeling is a tangent map method developed by Mikkola & Innanen (1999). This method computes the Lyapunov ex- ponents in an analytic way as a byproduct of the main numerical integration.21 _This

sort of numerical method, that allows analytic computations to be performed on the basis of numerical ones is called a semi-analytic method. The interest of this class of numerical method is that it involves a combination of analytic reasoning and numer- ical computing, in order to compute quantities not computable analytically to high accuracy. This sort of method, a hybrid of analytic and numerical methods, is quite common in practice and shows yet another way that a picture of the application of theory in terms of deductive inference is inadequate.

An interesting point made byWiegertet al.(2007) is that chaotic behaviour of this sort does not necessarily mean that the model breaks down entirely. Provided that over time scales longer than the Lyapunov time only the uncertainties of an object

within its orbit become large, then it is possible to study some of the properties of the object over long times, even getting fairly reliable determinations of quantities if they are stable across nearby orbits. This can be studied by numerically integrating an ensemble of objects with nearby initial conditions. Thus, the behaviour of an object

can in some circumstances be studied beyond its Lyapunov time scale at the expense of replacing a deterministic study by a statistical one. Thus, some degree of chaos, or nonlinearity, can be accommodated within the framework of celestial mechanics. Thus, feasible methods have ways of overcoming some of the limitations on knowledge imposed by nonlinearity.

Another interesting feature of this statistical approach is that it shows that by shifting how one thinks about getting information from the model, it can be possible to get useful information from the model even when it can no longer accomplish what it was intended to do. The double pendulum system provides a clearer example of this kind of phenomenon. It is primarily of interest as a theoretical and numerical study of chaotic behaviour in an easily tractable case. Accordingly, the ability to reliably and closely track trajectories is not the only, perhaps not even the central, concern. Even if we were interested in closely tracking the phase trajectory of a real double pendulum system, however, the strongly chaotic behaviour it exhibits under certain conditions would mean that we would have to abandon this aim in certain regions of phase space. In such a case, if we are to obtain useful information from the model we have to ask different questions (Corless, 1994a). And which questions we can usefully ask are ones where the answers are determined by stable quantities.

A chaotic dynamical system is characterized by its dynamical instability, insta- bility determined by extreme sensitivity of the dynamics to variation. If the initial conditions of a chaotic dynamical system are changed slightly, nearby trajectories will diverge at an exponential rate. But, such a dynamical system is also unstable in another sense: if you were to slightly perturb the dynamics (vector field) of the sys- tem you would also see exponential divergence of nearby trajectories. This is to say that the dynamics is unstable both in terms of small changes of the initial conditions and in terms of small changes of the dynamics. The reason this matters is that both the analytic model reduction techniques and the numerical solution methods involve variations of the initial conditions and/or the model (vector field). Accordingly, only quantities of the target system that are stable under such variations can be feasibly studied using mathematical modeling.

This, then, leads to the question as to what quantities are stable across variations of the model. One likely candidate for a stable quantity over variations of the model

are the Lyapunov exponents. Another is the regions of phase space where the dy- namics is chaotic, in the sense that we expect that conditions under which the system becomes chaotic will not change significantly with a small change in the model (vector field). It is for this reason that quite simplified or idealized models, provided that the simplifications or idealizations are not too severe (in an imprecise sense), can still give us useful information about the chaotic behaviour of a real double pendulum system, even when solved numerically. The stability of quantities over variations of the model generally requires proof, but even when proofs are not available evidence

that a quantity is stable can be obtained by considering an ensemble of nearby models and computing the same quantities to see if they are stable. In this way, a statis- tical study of the chaotic behaviour of the double pendulum system can be carried out, quite analogously to the study of solar system objects beyond their Lyapunov time scale. Since, classically, models are typically judged in terms of the accuracy of their solutions, chaotic dynamics would in this case be regarded as a breakdown of the model. What this technique of strategically changing the question shows is that looking to quantities that are stable under quite a broad range of variations allows a model that would be viewed as useless from a classical perspective to give useful information about the phenomenon.

Before moving on, it is worth mentioning that another important technique used to study nonlinear dynamical systems is linearization, which we have already seen illustrated above on the simple pendulum system. Often the dynamical behaviour of a nonlinear system can be studied over small (in a sense that is relative to the degree of nonlinearity)22 _{distances in time and space. This is effectively what is}

done by tracking a NEO within its Lyapunov time. A useful way of studying this analytically, however, is to expand the vector field in a Taylor series and to retain only the linear term. This yields a linear system which is much more amenable to formal analysis. Thus, a nonlinear system can often be replaced by a linear one at the expense of having a model that is only locally valid in phase space (near the point where the vector field is truncated). Linearization is a very important technique and it finds quite widespread usage as a means of local analysis. It is also a kind of

22_{This can be measured roughly by the Lipschitz constant of the vector field (see e.g., Henrici,}

model reduction technique, since linearizing the vector field changes the model from a nonlinear one to a linear one.

There are a wide variety of techniques in applied mathematics that follow the basic algebraic approach of expanding some nonlinear quantity in a power series and retaining only up to the linear term. The linear stability analysis of dynamical systems, for example, uses this technique, as does the standard analytic technique for the computation of the Lyapunov exponents of a dynamical system. So, indeed, do the renormalization group methods used in statistical mechanics for the computation of critical exponents of a system near its critical point, a case considered in detail by

Batterman (1998, 2002b). Geometrically speaking, this technique of linearization is useful anywhere where over some limited region a curved quantity can be accurately approximated by a flat one. We will see a variety of kinds of linear methods when we consider the construction of algorithms used in NEO modeling in the following subsection and in the algorithms for computing impact probability in chapter 5.