1.2.2 Nonlinear Dynamics

The dynamical systems of interest in chaos studies are nonlinear, such as the Lorenz model equations for convection in fluids:

(Lorenz)
dx
dt
= −σx+σy;

dy
dt
= −xz+rz−y;

dz
dt
= xy+bz

A dynamical system is characterized as linear or nonlinear depending on the nature of the equations of motion describing the target system. For concreteness, consider a differential equation system, such as dx ⁄ dt = Fx for a set of variables x = x1, x2,…, xn. These variables might represent positions, momenta, chemical concentration or other key features of the target system, and the system of equations tells us how these key variables change with time. Suppose that x1(t) and x2(t) are solutions of the equation system dx ⁄ dt = Fx. If the system of equations is linear, it can easily be shown that x3(t) = ax1(t) + bx2(t) is also a solution, where a and b are constants. This is known as the principle of linear superposition. So if the matrix of coefficients F does not contain any of the variables x or functions of them, then the principle of linear superposition holds. If the principle of linear superposition holds, then, roughly, a system behaves linearly if any multiplicative change in a variable, by a factor a say, implies a multiplicative or proportional change of its output by a. For example, if you start with your stereo at low volume and turn the volume control a little bit, the volume increases a little bit. If you now turn the control twice as far, the volume increases twice as much. This is an example of a linear response. In a nonlinear system, such as (Lorenz), linear superposition fails and a system need not change proportionally to the change in a variable. If you turn your volume control too far, the volume may not only increase more than the amount of turn, but whistles and various other distortions occur in the sound. These are examples of a nonlinear response.
1.2.3 State Space and the Faithful Model Assumption

Much of the modeling of physical systems takes place in what is called state space, an abstract mathematical space of points where each point represents a possible state of the system. An instantaneous state is taken to be characterized by the instantaneous values of the variables considered crucial for a complete description of the state. One advantage of working in state space is that it often allows us to study the geometric properties of the trajectories of the target system without knowing the exact solutions to the dynamical equations. When the state of the system is fully characterized by position and momentum variables, the resulting space is often called phase space. A model can be studied in state space by following its trajectory from the initial state to some chosen final state. The evolution equations govern the path—the history of state transitions—of the system in state space.

However, note that some crucial assumptions are being made here. We are assuming, for example, that a state of a system is characterized by the values of the crucial variables and that a physical state corresponds via these values to a point in state space. These assumptions allow us to develop mathematical models for the evolution of these points in state space and such models are taken to represent (perhaps through an isomorphism or some more complicated relation) the physical systems of interest. In other words, we assume that our mathematical models are faithful representations of physical systems and that the state spaces employed faithfully represent the space of actual possibilities of target systems. This package of assumptions is known as the faithful model assumption (e.g., Bishop 2005, 2006), and, in its idealized limit—the perfect model scenario—it can license the (perhaps sloppy) slide between model talk and system talk (i.e., whatever is true of the model is also true of the target system and vice versa). In the context of nonlinear models, faithfulness appears to be inadequate (§3).
1.2.4 Qualitative Definitions of Chaos

The question of defining chaos is basically the question what makes a dynamical system like (1) chaotic rather than nonchaotic. Stephen Kellert defines chaos theory as “the qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems” (1993, p. 2). This definition restricts chaos to being a property of nonlinear dynamical systems (although in his (1993), Kellert is sometimes ambiguous as to whether chaos is only a behavior of mathematical models or of actual real-world systems). That is, chaos is chiefly a property of particular types of mathematical models. Furthermore, Kellert's definition picks out two key features that are simultaneously present: instability and aperiodicity. Unstable systems are those exhibiting SDIC. Aperiodic behavior means that the system variables never repeat values in any regular fashion. I take it that the “theory” part of his definition has much to do with the “qualitative study” of such systems, so we'll leave that part for §2. Chaos, then, appears to be unstable aperiodic behavior in nonlinear dynamical systems.

This definition is both qualitative and restrictive. It is qualitative in that there are no mathematically precise criteria given for the unstable and aperiodic nature of the behavior in question, although there are some ways of characterizing these aspects (the notions of dynamical system and nonlinearity have precise mathematical meanings). Of course can one add mathematically precise definitions of instability and aperiodicity, but this precision may not actually lead to useful improvements in the definition (see below).

The definition is restrictive in that it limits chaos to be a property of mathematical models, so the import for real physical systems becomes tenuous. At this point we must invoke the faithful model assumption—namely, that our mathematical models and their state spaces have a close correspondence to target systems and their possible behaviors—to forge a link between this definition and chaos in real systems. Immediately we face two related questions here:

1. How faithful are our models? How strong is the correspondence with target systems? This relates to issues in realism and explanation (§5) as well as confirmation (§3).
2. Do features of our mathematical analyses, e.g., characterizations of instability, turn out to be oversimplified or problematic, such that their application to physical systems may not be useful?

Furthermore, Kellert's definition may also be too broad to pick out only chaotic behaviors. For instance, take the iterative map xn + 1 = cxn. This map obviously exhibits only orbits that are unstable and aperiodic. For instance, choosing the values c = 1.1 and x0 = .5, successive iterations will continue to increase and never return near the original value of x0. So Kellert's definition would classify this map as chaotic, but the map does not have any other properties qualifying it as chaotic. This suggests Kellert's definition of chaos would pick out a much broader set of behaviors than what is normally accepted as chaotic.

Part of Robert Batterman's (1993) discusses problematic definitions of chaos, namely, those that focus on notions of unpredictability. This certainly is neither necessary nor sufficient to distinguish chaos from sheer random behavior. Batterman does not actually specify an alternative definition of chaos. He suggests that exponential instability—the exponential divergence of two trajectories issuing forth from neighboring initial conditions—is a necessary condition, but leaves it open as to whether it is sufficient.