17.11: Chaos

Essential to Know

• Some interactions in natural systems are linear (or almost linear over the range of values present in the environment), but many are nonlinear because the components do not vary in direct proportion to each other. For example, doubling the food supply may cause the population of an animal species to double if food is scarce, but it may result in little change in population if food is already abundant.
• Linear systems are predictable. If we increase one parameter (e.g., the available food supply), the system will adjust to equilibrium (the population will rise to a new stable value that is related in direct proportion to the new food supply).
• Nonlinear systems behave unpredictably. Changing one parameter may lead to a new equilibrium (which may be drastically different from the original system) or may cause the system to oscillate in a chaotic way.
• Complex nonlinear systems with many interactions are chaotic and unpredictable, but they tend to vary irregularly within an identifiable range of conditions. For example, weather is inherently unpredictable, but climate (range and average conditions) does not generally change from year to year.
• Complex nonlinear systems may respond to unusual changes in one or more parameters by “jumping” to oscillate around a different set of average conditions.
• Natural systems include many interactions, at least some of which are nonlinear. All  Earth environmental processes including, atmosphere, and ocean processes are therefore chaotic.
• Because natural systems are chaotic, their exact state cannot ever be forecasted (predicted) accurately. These systems are inherently variable, and they may undergo sudden changes to new average conditions.

Understanding the Concepts

In most cases, we conceptualize the world around us as behaving according to simple rules. For example, it seems reasonable that if we doubled the temperature difference between ocean water and the overlying air, we would find that the rate of heat transfer between the two also doubled; or if we doubled the rate of food production, we would find that the animal species eating this food doubled in population. These are linear relationships in which one parameter changes in direct proportion to another. To be linear, relationships need not be one-to-one (as just described). The only requirement is that a certain amount of change in one parameter will always result in the change of another parameter by the same amount, even if one parameter increases and the other decreases (Fig. CC11-1).

Figure CC11-1. Linear relationships can be depicted by the straight line on a graph that plots one variable (x) against another (y). The value of one variable may (a) increase with an increase in the other variable, or (b) decrease with an increase in the other variable.

Linear relationships are simple to conceptualize and to express mathematically. They are also intuitively attractive. For example, the hypothesized linear effect of changing food production rate on a consumer population seems perfectly reasonable at first. However, it does not represent the true behavior of the systems. To understand why, think about what would happen if we were to drastically increase or decrease the food supply for an animal species that has no predators. If we drastically increased the food supply, the population would increase, but at some level, the species would become so crowded that some other factor (e.g., breathing space) would prevent it from increasing further. If we drastically decreased the food supply, the population would decline. Eventually, it would reach such a low level that the species would not be able to reproduce successfully because, for example, the survivors would have to spread so far apart to obtain sufficient food that there would be too few available mates. Because the hypothesized relationships do not maintain proportionality if the value of one parameter is changed too much, they are nonlinear.

Examples of nonlinear relationships are depicted in Figure CC11-2. Many, if not all, of the relationships that control the real world are nonlinear, which critically affects our ability to model the real world. Fortunately, many relationships, such as the one just discussed, may be linear or nearly so over a wide range of values of the two parameters (Fig. CC11-2). For instance, the food–population relationship discussed earlier may be nearly linear within the range of food production rates that are observed in the real world. In these circumstances, we can use linear relationships in our models and the models will be similar enough to the real world to be useful.

Figure CC11-2. Nonlinear relationships can be identified by the curvature of a graph that plots one variable (x) against another (y). Although the curvature can be complex, there is often a range of two values within which the relationship is approximately linear.

Not until the 1970s did scientists begin to examine the behavior of mathematical models that incorporate nonlinear relationships. What they found was startling. Models that use only linear relationships, no matter how complicated, always reach a new equilibrium if one of the parameters is changed (unless parameters are constantly changing). However, even the simplest of nonlinear models involving only two parameters may never reach an equilibrium. The system may oscillate in a seemingly random, or chaotic, way, never exactly repeating itself.

The mathematics of nonlinear systems can be complicated. Fortunately, we can understand chaotic behavior by considering a simple system called the “Lorenzian waterwheel,” named for Edward Lorenz, who first investigated the chaotic behavior of this system (Fig. CC11-3). In this waterwheel, the buckets are pivot-mounted so that they always stay upright (like the chairs in a Ferris wheel). Each bucket has a small hole through which water can escape (but not drip into a bucket below). This system can be modeled by three simple but nonlinear equations, and it has also been constructed and tested as an actual physical model. 

Figure CC11-3. The Lorenzian waterwheel. (a) If water is poured into the top bucket faster than it can escape from the bucket, the wheel will be set in motion. (b) If the water flow to the buckets is maintained, the wheel may settle into an equilibrium in which it rotates at constant speed. This can happen when, for example, the buckets are just empty by the time they reach the top and are refilled. (c) The acceleration and speed of the wheel depend on the amount of water in each bucket, which depends on the length of time that each bucket spends under the fill point, which depends on the speed of rotation of the wheel. These are nonlinear relationships, and the wheel may never settle into a constant rotation rate, even if the water input rate remains constant. Furthermore, the rotation rate may vary constantly, and the direction of rotation may even be reversed.

If we pour water into the top bucket, the waterwheel’s behavior depends on how fast the water is poured, how fast each bucket empties, and how quickly the wheel turns (which determines how long each bucket is in the right part of the rotation to be filled).

What do you think happens if we pour water into the Lorenzian waterwheel at different rates but leave everything else unchanged? If we pour the water in at a very slow rate, we might expect it to escape from the top bucket fast enough that the bucket does not fill and the waterwheel will not turn. This is indeed what happens. If we pour the water faster, the top bucket will partially fill and set the wheel in motion, bringing the next bucket under the pouring point to fill in turn and continue the rotation (Fig. CC11-3b). As each bucket moves through the rotation, it will progressively empty. We might expect that, if we maintain a stable rate of filling, the rotation will be smooth and the rate of rotation will increase as the fill rate increases. This is what happens when the fill rate is relatively slow. However, at faster fill rates something interesting happens.

Consider the wheel rotating relatively slowly. One or more successive buckets fills almost to the top as the wheel rotates. Because these buckets are heavy, they accelerate the rotation as they move toward the bottom point of the wheel, causing buckets behind them to fill for a shorter time. The full buckets can pass through the low point in the wheel before they have emptied, whereas the following buckets are filled less and empty completely. Once past the low point in the rotation, a once-full bucket (now partially full) may be heavier than the once partially filled buckets (now empty) behind it, so it now tends to slow the rotation of the wheel (Fig. CC11-3c).

We can see that the rotation rate of the wheel will vary constantly, and the direction of rotation may even reverse. However, what is most surprising about the motion is that, even though we do not vary the fill rate, the wheel’s rotation does not, as we might intuitively expect, settle into a regular back-and-forth oscillation. Instead, the rotation varies continuously back and forth in a pattern that is never repeated, a “chaotic” situation. This happens because of the nonlinearity in relationships, such as the relationship between the rate of rotation and the amount of filling of each bucket.

Because natural systems involve many nonlinear relationships, they can act like the Lorenzian waterwheel: sometimes they can appear stable or change smoothly to a new equilibrium if one of their important parameters changes, or they can oscillate chaotically. For example, a convection cell is similar to the waterwheel, although inverted. If heat is supplied very slowly to the lower part of a fluid and lost slowly from the top, the heat can be transferred by conduction within the fluid and there is no movement. At somewhat higher heating or cooling rates, smoothly flowing convection cells may form (CC3), and at even higher rates the convection cell motion becomes turbulent and chaotic.

Even seemingly simple biological systems can exhibit this type of behavior. For example, as discussed previously, populations respond in a nonlinear way to changes in food availability. If food supply increases, then population increases accordingly, but as population continues to increase, other factors eventually limit and reduce the population. Depending on the nature of these relationships, an increase in food supply can lead to a new, higher population of the consumer species, chaotic variations of this population, or even collapse and extinction if, for example, a small increase in population drastically reduces reproduction rates because of overcrowding.

Natural systems generally depend on many interactions, some of which are nonlinear. As a result, most natural systems are chaotic, and this fact has many implications for science. Since the 1970s, a new scientific subdiscipline has developed to study chaos. The term chaos has now generally been replaced by complexity. Complexity is often the preferred term because certain characteristics of complex nonlinear systems can be deduced that enable us to understand their behavior and how their component parameters might vary under certain circumstances, even though the precise future status of the system cannot be predicted. We do not need to fully understand the behavior of nonlinear systems unless we become research scientists, but several of their characteristics are important to all of us.

Nonlinear system models are extremely sensitive to very (infinitely) small differences in initial conditions. This characteristic was discovered by Edward Lorenz when he was using a computer model of weather patterns, and these observations are what started the systematic study of chaos. Lorenz found that if he started his mathematical model several times from the same set of input parameters (e.g., temperature and humidity distribution), the predicted weather exhibited variations that, although complicated, were reproduced exactly each time the model was run.

Lorenz then decided to start his model at a time interval later than the starting point of his previous model runs. He started this new set of calculations with the values of input parameters that his previous model runs had predicted would occur at this time interval. Unexpectedly, he found that his model quickly began to predict weather patterns that differed from those that the model had predicted for this same time period when the model had been started at the earlier time (Fig. CC11-4). The reason for this difference, Lorenz found, was that the nonlinear equations in his model generated completely different weather patterns when the later-start model run used input data that were rounded from the values produced in the earlier-start model runs. When he ran his model from the beginning, the computer maintained each parameter’s value at each time interval to six decimal places (e.g., 0.267902). When he started his model in the middle, he rounded the input data to three decimal places (e.g., 0.268).

Figure CC11-4. These two plot lines depict the results obtained from the same nonlinear mathematical model that simulates weather patterns. The only difference between the two model runs is that the starting-point data were rounded to a smaller number of decimal places for one model run than for the other. The patterns grow farther and farther apart until they lose any resemblance to each other. Many natural systems show this sensitive dependence on initial conditions. As a result, it is impossible to measure and model such systems well enough to predict their precise future behaviors.

This sensitive dependence on initial conditions has often been called the “butterfly effect.” This term originated from the example that a butterfly stirring the air one day in Beijing could alter the storm systems in New York the next month. As a consequence of the butterfly effect, models of nonlinear systems can never predict future behavior exactly. In addition, the further into the future a model predicts, the less likely it is to be accurate. For example, weather forecasts will never be much better than they are today. We will never be able to accurately predict the exact time or place a thunderstorm will occur on the next day, nor predict with certainty whether it will rain a few days in the future. Similarly, we will never be able to forecast or predict the exact future state of any complex nonlinear system, including plate tectonic movements, ecosystem dynamics, or fish and wildlife population variations.

Fortunately, although many complex nonlinear systems never reach equilibrium, they do tend to oscillate chaotically within a range of conditions (values of the component parameters) that models can predict. For example, weather is chaotic and unpredictable, but climate (the average and range of temperature, rainfall, etc., at a specific location and time of year) can be predicted with some confidence because it changes little from year to year. Hence, models of complex systems (CC10), if they adequately resemble the real world, can be used with reasonable success to predict the average conditions and range of variations that will occur in the future.

Unfortunately, although complex nonlinear systems do tend to oscillate chaotically within a predictable range of conditions, this dynamic “equilibrium” can be disturbed by small changes in component characteristics. In some instances, if a critical value of one or more components is changed, the system can “jump” from one set of average conditions to a completely different set of average conditions around which it oscillates chaotically. Thus, a system may appear to suddenly change drastically, even though none of the components of the system changed substantially before the jump occurred. For example, a very small increase in a parameter, such as the average temperature of the oceans and atmosphere, could have little effect on climate until a critical point is reached. At that point, the Earth’s climate could suddenly become much warmer or colder.  Recognition of this concept has led to the now well-known term “tipping point”.

We now know that this type of sudden climate change has occurred in the past (Chaps. 8, 9, Fig. 7-21). Similarly, sudden changes in ecosystems, such as drastic declines or blooms in certain species, may be a natural consequence of nature’s nonlinearity. Sudden changes in other complex systems, such as ocean circulation and the motions of tectonic plates, may have occurred in the past for this same reason. It is important to realize that sudden changes in natural systems have occurred in the past and will occur again in the future. Furthermore, no matter how well we model such systems, it will be very difficult to reproduce these changes faithfully, and virtually impossible to develop an accurate predictive capability that will alert us to them before they occur.