Chaos Theory: And Why It Matters

Posted by Keith Tidman

Computer-generated image demonstrating that the behaviour of dynamical systems is highly sensitive to initial conditions

Future events in a complex, dynamical, nonlinear system are determined by their initial conditions. In such cases, the dependence of events on initial conditions is highly sensitive. That exquisite sensitivity is capable of resulting in dramatically large differences in future outcomes and behaviours, depending on the actual initial conditions and their trajectory over time — how follow-on events nonlinearly cascade and unpredictably branch out along potentially myriad paths. The idea is at the heart of so-called ‘Chaos Theory’.

The effect may show up in a wide range of disciplines, including the natural, environmental, social, medical, and computer sciences (including artificial intelligence), mathematics and modeling, engineering — and philosophy — among others. The implication of sensitivity to initial conditions is that eventual, longer-term outcomes or events are largely unpredictable; however, that is not to say they are random — there’s an important difference. Chaos is not randomness; nor is it disorder*. There is no contradiction or inconsistency between chaos and determinism. Rather, there remains a cause-and-effect — that is, deterministic — relationship between those initial conditions and later events, even after the widening passage of time during which large nonlinear instabilities and disturbances expand exponentially. Effect becomes cause, cause becomes effect, which becomes cause . . . ad infinitum. As Chrysippus, a third-century BC Stoic philosopher, presciently remarked:

‘Everything that happens is followed by something else which depends on it by causal necessity. Likewise, everything that happens is preceded by something with which it is causally connected’.

Accordingly, the dynamical, nonlinear system’s future behaviour is completely determined by its initial conditions, even though the paths of the relationship — which quickly get massively complex via factors such as divergence, repetition, and feedback — may not be traceable. A corollary is that not just the future is unpredictable, but the past — history — also defies complete understanding and reconstruction, given the mind-boggling branching of events occurring over decades, centuries, and millennia. Our lives routinely demonstrate these principles: the long-term effects of initial conditions on complex, dynamical social, economic, ecologic, and pedagogic systems, to cite just a few examples, are likewise subject to chaos and unpredictability.

Chaos theory thus describes the behaviour of systems that are impossible to predict or control. These processes and phenomena have been described by the unique qualities of fractal patterns like the one above — graphically demonstrated, for example, by nerve pathways, sea shells, ferns, crystals, trees, stalagmites, rivers, snow flakes, canyons, lightning, peacocks, clouds, shorelines, and myriad other natural things. Fractal patterns, through their branching and recursive shape (repeated over and over), offer us a graphical, geometric image of chaos. They capture the infinite complexity of not just nature but of complex, nonlinear systems in general — including manmade ones, such as expanding cities and traffic patterns. Even tiny errors in measuring the state of a complex system get mega-amplified, making prediction unreliable, even impossible, in the longer term. In the words of the 20th-century physicist Richard Feynman:

‘Trying to understand the way nature works involves . . . beautiful tightropes of logic on which one has to walk in order not to make a mistake in predicting what will happen’.

The exquisite sensitivity to initial conditions is metaphorically described as the ‘butterfly effect’. The term was made famous by the mathematician and meteorologist Edward Lorenz in a 1972 paper in which he questioned whether the flapping of a butterfly’s wings in Brazil — an ostensibly miniscule change in initial conditions in space-time — might trigger a tornado in Texas — a massive consequential result stemming from the complexly intervening (unpredictable) sequence of events. As Aristotle foreshadowed, ‘The least initial deviation . . . is multiplied later a thousandfold’.

Lorenz’s work that accidentally led to this understanding and demonstration of chaos theory dated back to the preceding decade. In 1961 (in an era of limited computer power) he was performing a study of weather prediction, employing a computer model for his simulations. In wanting to run his simulation again, he rounded the variables from six to three digits, assuming that such an ever-so-tiny change couldn’t matter to the results — a commonsense expectation at the time. However, to the astonishment of Lorenz, the computer model resulted in weather predictions that radically differed from the first run — all the more so the longer the model ran using the slightly truncated initial conditions. This serendipitous event, though initially garnering little attention among Lorenz's academic peers, eventually ended up setting the stage for chaos theory.

Lorenz’s contributions came to qualify the classical laws of Nature represented by Isaac Newton, whose Mathematical Principles of Natural Philosophy three hundred-plus years earlier famously laid out a well-ordered, mechanical system — epically reducing the universe to ‘clockwork’ precision and predictability. It provided us, and still does, with a sufficiently workable approximation of the world we live in.

No allowance, in the preceding descriptions, for indeterminacy and unpredictability. That said, an important exception to determinism would require venturing beyond the macroscopic systems of the classical world into the microscopic systems of the quantum mechanical world — where indeterminism (probability) prevails. Today, some people construe the classical string of causes and effects and clockwork-like precision as perhaps pointing to an original cause in the form of some ultimate designer of the universe, or more simply a god — predetermining how the universe’s history is to unfold.

It is not the case, as has been thought too ambitiously by some, that all that humankind needs to do is get cleverer at acquiring deeper understanding, and dismiss any notion of limitations, in order to render everything predictable. Conforming to this reasoning, the 18th century Dutch thinker, Baruch Spinoza, asserted,

‘Nothing in Nature is random. . . . A thing appears random only through the incompleteness of our knowledge’.

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*Another example of chaos is brain activity, where a thought and the originating firing of neurons — among the staggering ninety billion neurons, one hundred trillion synapses, and unimaginable alternative pathways — results in the unpredictable, near-infinite sequence of electromechanical transmissions. Such exquisite goings-on may well have implications for consciousness and free will. Since consciousness is the root of self-identity — our own identity, and that of others — it matters that consciousness is simultaneously the product of, and subject to, the nonlinear complexity and unpredictability associated with chaos. The connections are embedded in realism. The saving grace is that cause-and-effect and determinism are, however, still in play in all possible permutations of how individual consciousness and the universe subtly connect.