The proximity of simplicity and chaos – On the essence of linear and non-linear dynamics

It is a nightmarefor pond owners: From one week to the other the water becomes cloudy and opaque, on the surface forms a green algae carpet, it stinks terribly, and the fish swim belly up. And yet just a short while agoeverything was okay with the water. What happened? The pond ‘tipped over’. ‘Tippingover’ describes a sudden oxygen deficiency in a body of water. As a consequence, all life in it depending on oxygen dies. This oxygen deficiency can occur when the degradation of dead algae and water plants consumes more oxygen than the living ones can produce. To put it bluntly, a last dead alga causes the oxygen concentration to drop below a critical value (this can also be caused by changes in weather conditions such as strong winds and rain), and fish and other living beings dependent on oxygen die. Because their bodies are now decomposed under oxygen consumption, the oxygen content falls even further. An irreversible process of dying has begun.

The suddenness of this process confuses many people. For ecological systems, they think of the same continuous and predictable processes, which they are accustomed to from everyday life. Here input (small change in the quantity of dead plants) and output (overall state of the pond) always share roughly the same proportion. This type of process we refer to as ‘linear’. Indeed the relatively simple (and linear) structure of the fundamental equations of physics, as found in Newton’s equation of mechanics, Maxwell’s equations of electromagnetism and Schrödinger’s and Dirac’s equations in quantum mechanics, respectively quantum field theory, suggests that the processes in nature can generally be well calculated and predicted. Physicists have therefore long believed that simplicity is the rule and nature is basically quite straight forward to describe and forecast. This may have been part of the reason for the natural scientists’ long upheld reductionist and mechanistic thinking about nature.

However, already the mathematicians of the 18th century recognized, that if one extends the two-body problem of gravity (the sun and a single moving planet, as described by Kepler’s laws) to merely three bodies, the resulting mathematical equations become very intricate(and Kepler’s laws do no longer hold exactly). And the dice game which had been familiar to people of the seventeenth century for many centuries shows that simple mechanical processes may not be predictable, because the initial conditions of the dice and their impact on the ground and wall are not (and cannot be) knownin sufficient detail. Meanwhile, we can generalize these findings: Many phenomena in nature do not correspond to the rather simple master cases of theoretical physics. With their many degrees of freedom (independent and therefore ‘freely selectable’ variables of movement in a system), they are so complex that they are anything but easily solvable and predictable. Contrary to the beliefs of the eighteenth century, nature, and especially man can hardly be described by means of the paradigm of an easily describable or even controllable machine.

Thus, already the system of three massive bodies mentioned above (‘three-body problem’) can display a rather strange behavior, in which even minimal changes to its initial conditions lead to large differences in its movements. In the 1970s and 1980s the physicists and their colleagues in mathematics discovered numerous physical systems and models that exhibited such curious characteristics. Supported by computer generated vivid representations of the underlying irregular motions these found a great deal of attention in the media and public. Thus, the “Mandelbrot set” representing such dynamics acquired some broader fame.

Today we call the characteristic of such systems “chaotic”. One of their important properties is that they are ‘non-linear’ (technically this means thatthe variables of the motion in the underlying (differential) equations occur in higher powers than one, i.e. the linear case). Unlike in colloquial terms, the word ‘chaos’ in physics does not characterize a presently untidy state of a system, but a characteristic of the dynamics of its temporal behavior: although the underlying laws unambiguously determine the behavior of a chaotic system, it is nevertheless irregular and unpredictable. Physicists therefore speak of ‘deterministic chaos’.

Such a connection between ‘determinism’ and ‘chaos’ may seem surprising at first sight. As a matter of fact, the term ‘deterministic’ is usually understood as ‘causally determined’. However, on distinguishing two different kinds of causal determination, this contradiction can be resolved: causal relationships can be of linear or non-linear kind. Linear relationships not only mean “same causes, same effects”, but also “similar causes, similar effects”, which makes the system well predictable over longer periods of time. This is no longer the case in the case of non-linear relationships: similar, very similar, indeed almost indistinguishable causes can here have very different effects. The principle “exactly the same causes, exactly the same effects” continues to hold, but no more does the principle “similar causes have similar effects”. Even small deviations from the exactness can cause great differences in the temporal dynamics of the system. Since the initial conditions can never be exactly defined or established in reality, the development of such a system is no longer predictable.

In (deterministically) chaotic systems at certain critical thresholds, so-called ‘tipping points’, it can happen that the entire system changes dramatically with only a minimal change in certain of its variables, whereas beyond these tipping points the systems behaves absolutely ‘normally’. This is what happened in a pond that tipped over: it changed into a completely different (in this case, dead) state. Hence, the basic philosophical assumption of Leibniz-and with that, in essenceof a very large part of the entire Western natural philosophy since Aristotle, that “nature makes no leaps” (“naturanon facitsaltus”[1]) is false. An important example is our global climate which has many such tipping points we know of today.

This insight holds some political explosiveness: Many people, including numerous politicians, unfortunately do not recognize that the characteristics of the climate on our planet with its countless non-linear feedback loopsis completely different from the usual linear trends according to which political and social developments in their eyes are to take shape. Social scientists (especially in the field of finance and financial markets) today refer to possible, but very unlikely abrupt, and often devastating events using the term “black swan” – a metaphor originally introduced by the philosophers John Stuart Mill and Karl Popper[2]. While the physics (and even before them the biologists) have recognized the characteristics of the complex and unpredictable dynamics of non-linearsystems, their colleagues from the social science (and especially the economists) all too often continue to use models which are simplified to the level of neglect, instead of pursuing an understanding of the more complex origins of, for example, “black swan” events. Hence,supervisory bodies continue to follow the paltry mathematical models of the financial industry and the recurring safety incantations of the nuclear lobby. But events like the 2008 capital markets crisis and the nuclear disaster in Fukushima in March 2011, and on larger scale the dynamics of the global climate under the greenhouse effect, demonstrate to us: We are dancing on a volcano. Accordingly, we must request clear information and take responsible actions. Contrary to political good will there is no lack of scientific insight and applicable mathematical models for that purpose. Non-linear processes have long received their merited attention in the natural sciences.

[1]This assumption which we find already with the Pre-SocrateHeraklit („PantaRhei“ – „Everything flows “), originates in this form with the natural researcher  Carl vonLinnéand constituted the biological as well as geological theory of gradualism. However, the subject of continuity of natural developments has always given rise to hefty discussion in the history of thought.

[2]Which a last, however, goes back to the Roman satirist Juvenal, as he stated that a loyal wife is a “raraavis in terris, nigroquesimillimacygno (a rare bird in all countries, most like a black swan).