Jonathan Lawhead

“ That is, being wealthier than those around you might itself have a positive impact on your subjective utility--an impact that is at least partially independent of the absolute quantity of wealth you possess. ”

“ In particular, we’ll focus on two aspects of the global climate which, while neither necessary nor sufficient for high dynamical complexity in themselves, are characteristic of complex systems: the presence of non-linear feedback mechanisms, and the presence of chaotic behavior. We’ll think about what it means for a system to be chaotic, and how the presence of feedback mechanisms (which are represented as non-linearities in the mathematics describing the system’s behavior) can contribute to chaos. ”

“ It’s easy to see that the story’s outcome given this behavior will be far different than the outcome in which their affections are purely driven by mutually reinforcing positive feedback loops. Rather than growing without bound, their affections will tend to stabilize at a particular level, the precise nature of which is determined by two factors: the initial conditions (how much they like each other to begin with) , and the level of responsiveness by each teen (how much Juliet’s affection responds to Romeo’s reciprocity, and how much Romeo’s affection responds to Juliet’s enthusiasm) . ”

“ I will argue that while “science by simulation” is an absolutely indispensable approach that climate science must take advantage of, it also comes with its own set of novel pitfalls, which must be carefully marked if they are to be avoided. More specifically, I argue that careful attention to the nature of chaos should force us to attend to the limitations of science by simulation, even in ideal conditions. ”

“ After all, the choice of a state space in which to represent a system is just a choice of how to describe that system, and so to notice that a system’s behavior is constrained in one space is just to notice that the system’s behavior is constrained period, though the degree of constraint can vary. ”

“ We’ve seen that state space trajectories in chaotic region diverge from one another, but we’ve said nothing at all about how quickly that divergence happens. As you might expect, this is a feature that varies from system to system: not all chaotic behavior is created equal. The rate of divergence between two trajectories is given by a particular number—the Lyapunov exponent—that varies from system to system (and from trajectory to trajectory within the system [23] ) . ”

“ I argued that while the classical scientific method of decomposing systems into their constituent parts and studying the behavior of those parts in isolation has been spectacularly successful in the history of science, a number of contemporary problems have forced us to look for tools to supplement that approach. We saw that both biology and climate science have begun to explore more holistic models, with the hope that those perspectives will shed some light on issues that have stymied the decompositionalist approach. ”

“ Similarly, it is possible for systems to find their way out of chaotic behavior. Attempting to simply divide systems into chaotic and non-chaotic groups drastically over-simplifies things, and obscures the importance of finding predictors of chaos—signs that a system may be approaching a chaotic region of its state space before it actually gets there [22] .
Another basic issue worth highlighting is that chaos has absolutely nothing to do with indeterminism: a chaotic system can be deterministic or stochastic, according to its underlying dynamics. ”

“ The failure of variations in utility to be tied linearly to variations in wealth, though, can be understood as a failure of condition (2) from Section 5.1.1—the wealth/utility relationship is like the Olympic treadmill. More recently, empirical work in the social sciences has gone even further. Kahneman and Deaton (2010) argue that utility (or, as they put it, “emotional well-being”) increases with the logarithm of wealth, but only up to a point. On their account, plotting the relationship between utility and wealth yields a strongly concave function, which is what we ought to expect. ”

“ Dynamically complex systems are (by definition) those which admit of sensible and useful consideration from a large variety of different perspectives; many interesting dynamically complex systems might exhibit chaotic behavior from some perspectives but not others. We should resist the temptation to even consider the question of whether systems like that are “really” chaotic or not in just the same way that we should resist the temptation to generally privilege one set of real patterns describing a system’s time-evolution over the others. ”

“ Positive feedback mechanisms are perhaps best described as “runaway” mechanisms; unless they’re checked (either by other mechanisms that are part of the system itself or by a change in input from the system’s environment) , they will tend to increase the value of some parameter of the system without limit. In the case of Romeo and Juliet, it’s easy to see that once the cycle is started, the romantic feelings that each of them has toward the other will, if left unchecked, grow without bound. ”

“ Just as with the Olympian treadmill we described above, the values of the relevant variables in the system of differential equations describing the system depend on one another in non-trivial ways; in the case of the treadmill, the value of a button-press varies with (and affects) the speed of the belt, and in the case of the logistic equation, the rate of population growth varies with (and affects) extant population. This general behavior—the presence of feedbacks—is characteristic of non-linear systems. ”

“ If the differential equations defining the system’s path through its state space contain no probabilistic elements, then the system will be deterministic. Many (most?) chaotic systems of scientific interest are deterministic. The confusion here stems from the observation that the behavior of systems in chaotic regions of their state space can be difficult to predict over significant time-scales, but this is not at all the same as their being non-deterministic. ”

“ Climate sensitivity, recall, is the relationship between the change in the global concentration of greenhouse gases (given in units of CO2-equivalent impacts on radiative forcings) and the change in the annual mean surface air temperature (see Chapter Four) . If the Earth were a simple system, free of feedbacks and other non-linearly interacting processes, this sensitivity would be a straightforwardly linear one: each doubling of CO2-e concentration would result in an increase of ~.30 , which would correspond to a mean surface temperature change of 1.2 degrees C at equilibrium [36] . ”

“ That is, neither non-linearity nor chaos (nor the conjunction of the two) is sufficient for dynamical complexity [2] .
5.1.1 Non-Linearity
Before we can tackle what it means to say that a system’s behavior is non-linear, we need to get some basic terminology under our belt. Complex systems theory is built largely on the back of a more general approach to scientific modeling called dynamical systems theory, which deals with the creation of mathematical models describing change (“dynamics”) in parts of the world (“systems”) as time progresses. ”

“ If our system is chaotic, then no matter how small we make our region the trajectories followed by the states that are included in it will, given enough time, diverge significantly [25] .
How much does this behavior actually limit the practice of predicting what chaotic systems will do in the future? Let’s keep exploring the mathematics and see what we can learn. ”

“ To put the point another way, restricting our attention to the behavior of some system considered as a collection of states can distract us from relevant factors in predicting the future of the system. ”

“ Feedback mechanisms are a very common way for natural systems to regulate their own behavior. Dynamically complex systems, with their layers of interlocking constraints, have ample opportunity to develop a tangled thicket of feedback loops. Jay Forrester, in his 1969 textbook on the prospects for developing computational models of city growth, writes that “a complex system is not a simple feedback loop where one system state dominates the behavior. ”

“ If the amount of money that our gambler should be willing to pay to play a game is constrained only by the demand that it be less than the expected return from the game, then this suggests that she should pay any finite amount of money for a chance to play the game just once. ”

“ If we add in enough initial energy, the angular velocity will be high enough that, rather than coming to rest at the apex of its swing toward the other side and dropping back down, the pendulum will continue on and spin over the top, something most schoolchildren have tried to do on playground swings. Depending on the initial push given, this over-the-top spin may happen only once, or it may happen several times. Eventually though, the behavior of the pendulum will decay back down into the class of trajectories depicted here, an event known as a phase change. ”

“ This powerful modeling tool is the heart of DyST.
Some systems can be modeled by a special class of differential equations: linear differential equations. Intuitively, a system’s behavior can be modeled by a set of linear differential equations if: (1) the behavior of the system is (in a sense that we shall articulate more precisely soon) the sum of the behavior of the parts of the system, and (2) the variables in the model of the system vary with respect to one another at constant rates [6] . ”

“ While feedback mechanisms are not definitive of complex systems like the climate, they are frequently the sources of non-linear behavior in the natural world, and so are often found in real-world complex systems. ”

“ The central mathematical tool of DyST is an equation that describes how different physical quantities of a system (e.g. force, mass, and velocity in Newtonian physics; populations of predator animals and prey animals in ecology; presence and concentration of certain atmospheric chemicals and global temperature in climatology) vary in relation to one another over time. ”

“ Romeo, in this story, likes to “play hard to get:” the more he sees that Juliet’s affections for him are growing, the less interested he is in her. Juliet, on the other hand, is responsive to encouragement: the more Romeo seems to like her, the more she likes him. ”

“ In a feedback-free climate system, we can parameterize 5 (n) such that , and such that . That is, we can assume that the net impact of positive and negative feedbacks on the total radiative flux is both constant and non-existent. However, just as with albedo, observations suggest that this simplification is inaccurate; . Discerning the value of is one of the most challenging (and important) tasks in contemporary climate modeling.
The presence of so many interacting feedback mechanisms is one of the features that makes climatology such a difficulty science to get right. ”

“ This is what it means for a system to behave chaotically: small differences in initial condition are magnified into larger differences as the system evolves, so trajectories that start very close together in state space need not stay close together. ”

“ Given the its central role in the global carbon cycle, any feedback mechanism that negatively impacts the ocean’s ability to act as a carbon sink is likely to make an appreciable difference to the future of the climate in general. ”

“ In chaotic systems, the distance between two trajectories through the state space of the system increases exponentially as time goes by—two states that start off very close together will eventually evolve into states that are quite far apart. How quickly this divergence takes place is captured by the value of the Lyapunov exponent for the trajectories under consideration (with the “worst-case” rate of divergence defining the MLE) . Generalizing from particular pairs of trajectories, we can think about defining a region in the state space. ”

“ Equivalently, if the size of the neighborhood is zero—if the neighborhood includes one and only one point—then we can be sure of the system’s position in its state space at any later time (assuming no stochasticity in our equations) . This is why the point that chaotic dynamics are not the same thing as indeterministic dynamics is so important. However:
As the Lyapunov exponent approaches zero, the second term on the right side of the inequality in 5 (j) approaches unity. ”

“ In Section 5.2, we will discuss one of the defining tools in the climatologist’s tool-kit: computer simulation. The construction of simulations—computer-solved models designed to be run repeatedly—is a methodological innovation common to many complex system sciences; we’ll think about why this is the case, and consider the relationship between the challenges presented by non-linearity and chaos, and the unprecedented methodological opportunities presented by modern supercomputers. ”

“ Angular velocity always falls as angular displacement grows (and vice-versa) , so each full period will look like an ellipse, and the loss of energy to friction will mean that each period will be represented by a slightly smaller ellipse as the system spirals toward its equilibrium position of zero displacement and zero velocity: straight up and down, and not moving. See Figure 5.1 for a rough plot of what the graph of this situation would look like in a state-space for the pendulum. ”

“ It seems reasonable, that is, to think that getting $10 would not only leave you with more utility--make you happier--than getting $5 would, but also that it would leave you with twice as much utility. Empirical investigation has not supported this idea, though, and contemporary economic theory generally holds that the relationship between wealth and utility is non-linear. ”

“ More recently, the non-classical revolution in the early 20th century forced us to refine concepts that we’d taken to be rock-solid in our conception of the world, and the philosophical implications of quantum mechanics and relativity are still being fought out in ways that are actually relevant to the progress of science. ”