23.6 23.5 Dynamical Systems and Chaos
In this and the two preceding chapters, we have considered systems that change over time: bank accounts, the amount due on a loan, and the size of a population. Other examples are a dripping faucet, a playground swing, a pinball play, the solar system, the business cycle, epidemics, the passage of a drug through the human body, and the weather. Some of these systems are very predictable (interest on a bank account), while others are notoriously unpredictable (the weather). Some involve no outside influences (the amount due on a loan, assuming that you don’t get behind on payments!), while others are the result of many contributing factors (the business cycle).
In some systems (such as the population of a country), the state of the system may depend largely on its states at previous times (e.g., last year’s population), while in other systems (such as an epidemic) chance may play a large role (e.g., in who and how many become infected). We are interested in modeling systems, such as a fishery, as they operate without influence from outside or from chance. The applicable mathematical tool is a dynamical system.
Dynamical System DEFINITION
A dynamical system is a mathematical model for a system whose state evolves with time and whose future states depend deterministically on its present and past states.
To make this definition meaningful, we need to be explicit about what is meant by deterministic.
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Deterministic DEFINITION
A system is deterministic if its changes through time depend only on natural and mathematical laws and are not substantially affected by what we consider to be chance or free will.
An example of a deterministic system is the path of a golf putt, which is governed by gravity, terrain, wind, and the force imparted by the golfer. A nonexample is the outcome of a vigorous toss of a coin or a random number generator; although the result, like the golf putt, is determined by physical laws, we consider the result to be random. Another nonexample is the outcome of an election, which involves choices by humans.
Mathematical Chaos
We think of chaos as referring to general confusion, unpredictability, and apparent randomness. Mathematicians and other scientists use the word to describe systems whose behavior over time is inherently unpredictable.
Chaos DEFINITION
A dynamical system exhibits chaos if it is:
- Orbitally dense—any state is near one that eventually will recur
- Transitive—from any state you can eventually get close to any other
- Sensitive—a small change in the initial state can produce widely diverging results later
EXAMPLE 11 Chaos in Manhattan
You may already know from experience that getting around Manhattan can be a chaotic experience, in the ordinary sense of the word. How is Manhattan’s subway system also chaotic in the mathematical sense?
- Orbitally dense: Subway trains “orbit,” periodically visiting subway stops, and everybody lives near a subway stop.
- Transitive: You can get close to anywhere else in Manhattan by taking the subway.
- Sensitive: If you get on the wrong train, you could wind up miles from where you want to be.
Because the system covers the island of Manhattan, #1 is actually a consequence of #2. Also, anyone who has gotten on the wrong subway train or bus realizes that #3 is an inevitable consequence of #1 and #2. So, in fact, #1 and #3 both follow from #2—a conclusion that is true not just of this Manhattan example but of a large class of dynamical systems.
Self Check 9
Why does #1 follow from #2 for the Manhattan subway system?
- Just go to a subway stop and wait for a train that (via connections) goes to where you want to go.
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The most noticeable property of a chaotic system is sensitivity—that a small change now can make a big difference later.
This feature is sometimes known as the butterfly effect, from the title of a 1979 talk by meteorologist E. N. Lorenz (1917–2008): “Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?” (The phrase probably traces to a 1953 science fiction story by Ray Bradbury, “A Sound of Thunder,” in which history is changed by a time-traveler who steps on and kills a prehistoric butterfly.)
We can get a feel for chaotic systems by playing with some examples of an iterated function system (IFS). The fancy name just means that we take an initial value, apply a function to it, then repeat over and over. This is exactly what we did earlier, geometrically, with reproduction curves for populations. (See Section 19.5, pages 807-811, for more about IFS and their connection to fractals.)
Iterated function system (IFS) DEFINITION
An iterated function system (IFS) is a sequence of numbers or geometric objects in which each next element is produced from the previous one by applying a consistent function (rule) to the previous element.
EXAMPLE 12 Doubling on a “Stone Age” Calculator
Imagine that you have a calculator that keeps only the last two digits of a number. It has a special key marked [DBL] that doubles the number in the display and keeps only the last two digits. For example, [DBL] applied to 52 gives 04 (not 104). Let’s start with two numbers that are as close together as can be on this calculator, such as 37 and 38. As we push the [DBL] key over and over again, will the result stay close?
We get 37, 74, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52, 04, … 38, 76, 52, 04, 08, 16, 32, 64, 28, 56, 12, 24, 48, 96, … Already, by the fourth iteration, the two sequences are far apart. The function used in this iterated function system is
where the mod notation of modular arithmetic (introduced in Chapter 17 on page 715) means to take the remainder when 2x is divided by 100.
Self Check 10
What happens if you start with 01?
- After 02, you wind up eventually in the same loop as when starting with 37 or 38: 04, 08, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52, 04, …. It can be shown that the length of the loop (20) must divide the modulus (100).
EXAMPLE 13 The Solar System
The American moon landings in 1969 and later, as well as all other space missions, were possible because of the predictability, or determinism, of the solar system. The moon and planets follow their orbits like clockwork. So how could the solar system be chaotic?
Over tens of millions of years, the orbit of each planet is chaotic, meaning that the slightest change in its position or velocity—due to, say, a comet passing nearby—could produce a huge difference later. In mid-2015, we learned that at least two of Pluto’s five moons (Nix and Hydra, discovered only in 2005) rotate and tumble chaotically.
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More down-to-earth examples of physical systems that can exhibit chaos include the fluttering of a falling autumn leaf, heart arrhythmias, and the Tilt-A-Whirl amusement park ride.
Chaos in Biological Populations
If we measure this year’s population as a fraction of the carrying capacity, and do the same for next year’s population as a fraction , the logistic population model takes the form
where the Greek letter lambda is the amount by which the population is multiplied when the population is near 0. When expanded, the equation has the familiar form of a quadratic in :
Values of slighlty larger than 1 correspond to natural growth rates near 0, and the behavior we noted earlier, of the population gradually creeping toward the carrying capacity, takes place. But for larger values of , corresponding to high growth rates of the population, there are other outcomes, as the next example illustrates.
The population can more quickly expand and (especially) contract (as it gets close to the carrying capacity).
EXAMPLE 14 The Logistic Population Model
What behaviors can occur in the logistic model?
For different values of the parameter and different starting values for the population fraction, each of the behaviors of Figure 23.12 on page 966 can occur:
- and starting population fraction x = 0.357 produces the values 0. 357, 0.643, 0.643, …, the pattern shown in the lower graph in Figure 23.12a.
- and starting population fraction produces the values 0.235, 0. 557, 0.765, 0.558, 0.765, 0.558, …, the pattern shown in the lower graph in Figure 23.12b.
- and starting population fraction produces the values 0.550, 0. 619, 0.590, 0.605, 0.598, 0.601, 0.599, …, the pattern shown in the lower graph in Figure 23.12c.
In other words, for population growth rates (values of ) that are fairly close together (2.5, 2.8, 3.1), the population evolves very differently. This is a surprising and nonintuitive conclusion.
But there is more. For and any starting population fraction, the population does not settle down into any of the patterns of Figure 23.12; year after year, it wanders “unpredictably” all over the place (Figure 23.17). This is chaotic behavior: It is deterministic, complex, but—in the long run—unpredictable. In the short run, the behavior is completely predictable. For example, from this year’s population fraction, the equation tells us exactly what next year’s will be. Repeating the use of the equation, we can determine what it will be the following year. But as the years pass, any sense of pattern gets lost in the complexity.
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Self Check 11
Which pattern do you arrive at if you start with and starting population fraction ?
- The sequence goes 0.5 00, 0.675, 0.5 92, 0.65 2, 0.613, 641, …, and centers down on 1.7/2.7 ≈ 0.630.
The potentially chaotic behavior of a biological population is bad news for those who manage a ranch or any biological population in the wild or in captivity. In recent years, the lobster catch in Maine has been much higher than in previous years, reaching record levels, for no discernible reason. On the other hand, in the late 1950s, the annual harvest of Dungeness crabs off the central California coast declined from 12 million pounds to less than 1 million pounds without any evidence of disease, heightened predation, or increased crabbing effort. Researchers who modeled that population in 1994 found that booms and busts are the rule. The population can remain nearly level for generations before suddenly exploding or crashing without warning.
Searching for an environmental cause for these fluctuations could be futile, because there may not be one. Moreover, observing the population over a few generations provides no help in predicting future behavior.
EXAMPLE 15 Childhood Disease Epidemics
The incidence of childhood diseases such as chickenpox and measles varies greatly from year to year. Why?
There are four plausible explanations for the fluctuations:
- There is an underlying regular cycle that is perturbed and occasionally overwhelmed by random events (“noise”).
- There is no pattern, because the fluctuations are due solely to chance.
- There is no discernible pattern, because such fluctuations are inherent in the epidemiology of the disease, a chaotic system.
- Parents refuse to vaccinate their children against the disease, or the vaccine becomes less effective, thereby increasing the numbers of children susceptible.
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- The first explanation, a perturbed cycle, fits chickenpox, with a cycle of one year.
- For measles, either the second or the third explanation may be correct, depending on the size of the community. For small communities, chance is an adequate explanation. For large communities, historical data from before the era of mass immunization suggest that measles cases were chaotic. That doesn’t mean that they occurred at random, but rather that they were unpredictable. Research also shows that there is a critical community size above which a disease will not die out solely by chance. For measles, this size is about 250,000.
- The two factors in the fourth explanation seems to account for recent flare-ups in whooping cough.
EXAMPLE 16 Chaos in Your Laptop
Does a computer always do what it is supposed to? How could your laptop computer be chaotic?
The Intel i5 chip common in many computers has 731 million transistors. They are subject to physical laws and environmental conditions, and they don’t all always respond in the same amount of time to do their commanded tasks. Researchers have found that running the same program (of several billion instructions) repeatedly under identical conditions can take greatly varying amounts of time. However, for some programs, the pattern of times displays clear evidence of mathematical chaos.
What you need to understand about chaos is that behavior that appears to be random can be produced even with very simple systems that are completely governed by deterministic rules. Just because the behavior appears chaotic does not mean a lack of underlying order and structure, though discovering that structure may be difficult.
Even if we discover the structure, prediction may elude us because of chaos. If we had an absolutely correct model of how weather behaves and measurements at every location on and above Earth, we might still not be able to forecast the weather accurately a week ahead.
What about the fishery extinctions? Perhaps the fishers and the fish were victims not of greed or chance, but of the chaotic nature of the reproduction curve for the fish.