Chaos Theory: Understanding Randomness

Chaos is a word humans habitually use to describe the uncertain and unpredictable nature of the universe. And oftentimes, we are right. Chaos is a vessel through which events exhibit erratic behavior, but that is not the end of it. There is a profound truth in how organized systems tend to chaos and there is a simple beauty in how it even comes to be in this seemingly deterministic world. Let us begin by putting a definition next to the term chaos.

Fig 1. Lorenz Attractor. This shows the chaotic divergence of paths even for points that started close together.

Chaos is defined to be unpredictable behavior that owes great sensitivity to small changes in conditions. The first real exposure to chaos was attributed to Edward Lorenz in the 1960s. After examining weather conditions on different trials, he realized that the conditions were drastically different depending on how much the initial conditions changed (even if it was some slight decimal variation). His findings revealed that chaos is a phenomenon that was greatly sensitive to the initial conditions; no matter how close the starting conditions were, some external factors would make the result differ, making the whole model unpredictable and difficult to follow.

That means that the future cannot be foretold with great accuracy when it comes to chaotic systems. If you have great differences in outcome based on slight differences in initial conditions, the future becomes a blur. Since we do not have the means to get the most accurate starting point (down to the very final decimal), it gets increasingly more difficult to pinpoint the future the further away it is. That is the nature of chaos.

Are there chaotic systems around me?

So, where is chaos? Most people, when they envision chaos, see a myriad of phenomena intersecting and overlapping in berserk order. It is hard to envision chaos in a world that seemingly exudes order, but more often than not, chaos is found in the most unbelievable corners. Consider the kitchen faucet found in most households all around the world. The rhythm of a steadily dripping kitchen faucet gives way to higher rates until the rhythm becomes complex enough to be deemed unpredictable.

The time interval of each drop is not repeated in a steady order, but in patterns that cannot be attributed to a structure. That is one example of chaos. Another example includes the commonly known weather forecast, which quite rarely matches up to the right degree Celsius in advance. Other examples not as commonly known include our solar system itself (the motion of celestial bodies flinging out of orbit or colliding with others over time, the double pendulum, and the spread of diseases (including our recent foe, the Coronavirus).

Chaos vs. randomness

These events, all spiraling into chaos, may seem random because we associate randomness with unpredictability. However, there is a clear distinction between chaos and randomness. Even though chaotic systems can exhibit random behavior, both are not interlinked intimately enough to be considered the same. Pure randomness, on the one hand, does not depend on the initial conditions at all but rather generally follows the probability breakdown of the events. No matter what the initial conditions are, the outcome is not determined by it. Consider tossing a coin, for example. If you toss the coin a million times, it still does not guarantee that it will be a tails or a heads the next time.

However, on the other hand, chaos is quite dependent on the initial conditions as we saw previously. Even though a butterfly flapping its wing may randomly result in a tornado in Texas, if the exact initial conditions are mimicked again, we would witness the same outcome. The same is true for random number generators in computers. It uses an already-established algorithm that generates those random numbers, which means the output is not random but it is still chaotic. All of this points to the crucial fact that in a purely mathematical world where you can specify initial conditions exactly, chaotic systems are fully deterministic. Even though randomness can arise from determinism, randomness is not strictly defined by it, whereas chaos is.

That being said, even though chaotic systems are deterministic, there are challenges that people face when dealing with them. Chaos, more often than not, throws reliability out the window. Even very accurate measurements of the current state become futile indicators of the future. One has to measure the system again and again to find out where it is at present. That is why long-term behavior is nigh impossible to predict; chaotic systems such as the weather, which many people depend on daily, need to be estimated by a set of predictions of varying multiple initial conditions instead of using just one. Even so, trying to predict more than a month in advance is almost as reliable as a good guess.

Bright side of chaos

Fig 2. A bifurcation diagram showing how the population numbers split in chaotic ways as the growth rate increases.

However, chaos is not always just challenges. Sometimes, chaos brings forth patterns in important areas of reality that provide a holistic model in revealing the true nature of many complicated systems. Consider the population growth model. Starting with an initial population, multiplied by the growth rate and a constraint to keep the population within reasonable margins, the equation will start to settle in on a number that defines its equilibrium population. However, as you increase the growth rate, after a certain point, the population number destabilizes and oscillates from multiple equilibrium values. It does this for a while until it completely loses a pattern and bounces around at random to different population numbers. As the growth rate increases further still, order returns. This process then repeats. This mapping of the population values to time which depends on the previous values is called a logistic map.

What makes the study of the logistic map so important is not only that the organization of these periodic and chaotic regimes can be completely understood with simple tools, but that despite its simplicity it displays

how periodic and chaotic behavior are interlaced, bringing light to the

mechanisms responsible for the appearance of chaotic behavior. Studying these models gives solutions to generating randomness, monitoring and controlling the rhythm of heart rates, predicting traffic flows, and other important engineering applications, all out of deterministic machines. This means unpredictable and complicated phenomena are not completely beyond our reach.

Closing remarks

With all that being said, chaos is an integral part of reality that we cannot ignore. Even though it is greatly sensitive to initial conditions, it still allows us to glimpse at randomness in deterministic models of nature. It not only allows us to give structure to patterns where there does not seem to be any, but it also enables us to understand the chaotic systems that are so close to us. Starting from the swirling of our solar system to a dripping kitchen faucet, chaotic systems push us to rethink and redefine our limitations in measurements. There is truth in unpredictability, and by studying chaos closely, we can find it.