I saw that sign “Curva Peligrosa” on a highway from Mexico City to Cuernavaca. The young woman who was driving never slowed down on that mountain road. With the same carefree abandon, humanity is now hurtling ahead on the road of historical and environmental change. The young woman and I made it around that curve; but who knows?

There are curves, geometric curves, used in modelling natural and social systems that depict both dangers and opportunities. Some are called by disaster-evocative names like Catastrophe, others by names of calming beauty like Swallowtail and Butterfly, still others by cold scientific names like Logistic. There are equations of Chaos and equations of Apocalypse. Is there any connecting thread between them that we can discover? Are there clues that would help us steer around that dangerous curve?

Many growth curves look like the logistic or “lazy S” curve: a lower plateau, followed by a steep rise followed by an upper plateau.

Figure 1. Figure 2. Figure 3.

This depicts the growth of bacterial colonies, for example, or of animal or human populations. The first (rising) part of the curve is an exponential function, because geometric growth is occurring in an environment with plentiful resources. Then, after the inflection (slope change) in mid-rise, a levelling-off takes place, as the resources become scarce, and a balance is achieved at the upper plateau. This latter part of the logistic is an inverse exponential, or a logarithmic curve, approaching the upper limit asymptotically. (See Figure 1.)

If it should happen that the population growth overshoots the point of balance, resources may become totally exhausted and the population may crash, either partially till some lower balance is restored, or totally to extinction (perhaps only local if we are dealing with local populations). If a crash occurs, the curve is no longer a logistic, but has the shape of a bell curve, like the curve of normal distribution (though the equation may be different). This is illustrated in Figure 2. Or rather, it may be a parabola, which better corresponds to the quadratic equation given below. Or, as Glieck suggests in his book “Chaos”, any curve with a hump (a single maximum) would do.

The logistic curve is also observed in many chemical reactions, for instance an acid-base titration, in which the pH rises along such a curve when a base is added dropwise to an acid, until an indicator like litmus or phenolphthalein or bromothymol blue suddenly changes color near the inflection point.

In my articles “Pentagon of Peace” and “Pax Democratica”, I posit similar logistic curves for the growth of peace, justice and democracy in the world, from primitive through transitional to modern societies. In some cases, as in the case of peace, the values START high in primitive societies, then dip during transition, and are expected to rise again at full maturity. (Such a “dip curve”, an inverted parabola, is a negative derivative of the logistic. It looks like the “potential well” sometimes postulated in particle physics. It is shown in Figure 3.) A logistic progression was already shown to exist by Rostow in his “Stages of Economic Development”, and another such pattern is well known for population growth in the so-called “demographic transition”.

The pattern of plateaus interspersed with spurts of growth (like repeated logistics building on top of each other) is generalized in my essay Rise and Run, which compares it to a staircase; by Rudolph Rummel in his “Conflict Helix”; by Jay Gould in his theory of punctuated equilibrium in biological evolution; by many theorists of child development; and by Toynbee in his parable of the mountain climber (a metaphor for the succession of civilizations), to name only a few examples of this very general pattern.

The equation of the logistic curve as usually given is actually not in terms of exponential and logarithmic functions; according to Leo Starobin, it is as follows:

= X (a – bX) (1)

where t is time, X is the essential variable such as population or wealth etc., and the parameters a and b are positive constants.

I was struck by the similarity of equation (1) to the Chaos Equation, also written up by Starobin on another occasion. The Chaos Equation looks as follows:

X = aX (1 – X ) (2)

Here t+1 denotes the next time period or “generation” in an iteration procedure, in which the previously calculated value of X is substituted on the right hand side to give the next value of X on the left hand side, which is then in turn substituted in the next step, and so on. Initially we start with some arbitrary starting value of X.

As Starobin explains, when the value of a is between 1 and 3, X eventually (after many iterations) settles down to a steady value, which is (1 – 1/a), and then no longer changes with further iterations. This steady value is called the “attractor”. But when a is between 3 and 3.43, there are two attractors, one on each side of (1 – 1/a), and the value of X, after many iterations, comes to oscillate between them as the iteration proceeds. (This is similar to the situation described in my essay “The Key”, in which the solution of a first-degree differential equation can be either an exponential approaching a limit asymptotically, or a sine-wave oscillation which alternately overshoots that limit and comes back repeatedly like a pendulum or a vibrating string.)

This splitting of an attractor is called a “bifurcation”, which is not quite the same use of the term as in Thom’s Catastrophe Theory, both being somewhat different from its use in Prigogine’s theory of dissipative structures – but the three bifurcation concepts resonate together in evocative ways, and perhaps there is a deep connection.

Then, when the parameter a in equation (2) increases above 3.43, there are 4 alternating attractors; then 8, then 16…in progressively decreasing intervals of increases in a; i.e. the “higher harmonics” don’t last as long as the “fundamental tone”, to make an analogy to music. Finally, when a exceeds 3.57, there are an “infinite” number of attractors (whether it is merely very large or truly infinite is not clear), and the system enters a state called Chaos. Values of X now seem to vary randomly during successive iterations, though the system remains strictly deterministic, i.e. the successive values of X can be calculated from each previous value with complete certainty and accuracy.

After reading Starobin’s account of the Chaos equation, I tried it on the computer, and found that at a = 4 (and perhaps at other values of a as well), X goes suddenly and abruptly from a high value of X = 1 right down to X = 0, indicating a total crash to extinction if X denotes a population. Thereafter, of course, repeated iterations would yield nothing but a string of zeros. Since the parameter a is defined as r + 1, where r is the rate of population increase, this means that at a high rate of population increase, high enough to be in the Chaos region, sudden extinction is possible even when the population levels are very high.

The Chaos system is characterized by an inherent instability in which anything is possible. This is because it is extremely sensitive to even very small external influences; a very slight push can tip it over into a drastically different condition. With respect to the prediction of weather (which is a chaotic system), this has been called “the butterfly effect”: when a butterfly flutters its wings in Bangkok, this can bring on a storm in Kansas.

The successive bifurcations in the 2,4,8,16… mode are also discussed in the book called “Chaos” by James Glieck. The generality of this pattern has been demonstrated by Mitchell Feigenbaum in Los Alamos (of all places). It seems that these transition limits occur in many different types of equations (e.g. one containing the trigonometric function sine), and they all converge with the SAME ratio, 4.6692 etc. (an irrational number). Something in all these very different equations is SCALING, in a way often observed in fractal geometry; but surprisingly, scaling with the SAME constant, an irrational number perhaps as fundamental as pi or e or the Golden Mean.

Chaos is a non-linear system; it occurs for example in the turbulent flow of a liquid, after its transition from laminar (ordered, regular) flow at the Reynolds limit. Laminar flow occurs at lower flow velocities, turbulent flow at higher flow velocities. We can visualize intuitively how forcing the rate of flow beyond the limit causes the molecular structure of the liquid to lose its ability to adapt or accommodate to the strong forces acting upon it, and falls apart into a sort of panic. Yet turbulent flow is a very common feature in nature; images of pastoral quiet and beauty like a “babbling brook” is really a system in Chaos; so are those lovely breakers on the beach, or the wake behind a moving boat, or the white-water rapids that have been compared to the crises of our rapidly changing world that we have to navigate. (Cf. “The Rapids of Change” by Theobald.)

It is said (but I have not seen it shown) that, if the Chaos system is pushed even harder (like the liquid flow being forced to very high rates), it can go beyond turbulence and emerge with a new regularity, but this time with a periodicity of 3 rather than 2; i.e. the first splitting (trifurcation?) goes to 3 attractors, followed by 9, then 27, etc. and on to a new Chaos. Thus the 3 exponential series replaces the 2 one. Is there New Order beyond Chaos? (A very pertinent question in the contemporary world.) However, at very high growth or flow rates, sudden extinction may intervene before the New Trinity Order is achieved. (“Trinity” was also the code name given to the first nuclear explosion in Alamagordo in 1945.) In any case, the trifurcation sequence, if indeed it is real at all, would probably occur only for a very fleeting interval before being swallowed up in the next Chaos.

However, I want to go back to the similarity between the equation of the logistic curve and the Chaos equation, both presumably describing population growth, among other growth phenomena, but in different ways. My aim is to explore what the formal similarity means, if anything.

The right-hand sides of equations (1) and (2) become identical if a = b, i.e. if the logistic equation uses only one parameter rather than two. It would seem that (2) is only a special case of (1), perhaps used by Starobin only to make the calculations simpler. Perhaps the more general logistic equation (1) would also bifurcate and then go to Chaos when subjected to iteration.

However, the left-hand sides of equations (1) and (2) are not identical, because (1) is not iterated while (2) is. Also, equation (1) is more like a differential equation and (2) like a finite-difference equation, i.e. growth is seen as continuous in (1) and as discontinuous (making distinct jumps at “generations” in (2).

Now in our philosophical interpretations, wild as they may be, the logistic equation (1) is regarded as beneficial, an approach to a Utopian state of Peace. Justice, Freedom, Democracy, Wealth, and a stable population level. (Cf. “Pax Democratica”.), though Starobin warns that the logistic may become the bell curve and indicate a crash. But, if realized, the upper plateau of the logistic would represent the TRUE New World Order, which human world society could reach if only we could also arrive at an equilibrium with Nature. On the other hand, equation (2) may lead us to Chaos from a previous state of Order (a kind of a Creation in reverse), and the stable population level reached may be zero. Is this a “super-bifurcation” of the future? Is this “double exposure” to be expected in a time of crisis, when we face either breakthrough or breakdown?

Another interpretation is possible. The equations as a whole are really quite dissimilar, from two entirely different worlds of mathematics. Equation (1) describes the graph whose slope is a quadratic function (i.e. a parabola with its hump vertically upward). Equation (2), on the other hand, does not describe a graph at all; it is a formula for iteration. Chaos comes only with iteration. There is no trace of Chaos in graphs. In equations (1) and (2), the similarity of the right-hand sides is less important than the dissimilarity of the left-hand sides, which define the operations to be performed – either graphing or iteration.

Now let us explore another aspect of our “dangerous curves”. I am referring to another class of curves altogether, coming from Thom’s Catastrophe Theory, as described in an article by Christopher Zeeman (Scientific American, April 1976). These come from yet another branch of mathematics, namely topology. Yet again, by rather far-fetched playing around with curve shapes, I want to look for relationships. Blending Chaos and Catastrophe appeals to my perverse nature.

Thom proved that there are only 7 “elementary catastrophes”, and he discovered and described all of them. The technical definition of a “catastrophe” is a curve or a surface or a space or a higher-dimensional entity in which the dependent (behaviour) variable or variables show a discontinuity even though the several independent (control) variables are continuous. Why should the juicy word “catastrophe” be used for such an abstruse concept? Well, the curves, surfaces etc. show sudden jumps which model various abrupt changes in behaviour (like a dog turning from fearful withdrawal to raging attack) or in ideology (like a fascist suddenly flipping into a communist, or a Christian being converted into a Muslim) or in economics (from a bear market to a bull market) or in eating habits (from starving to gorging in an anorexic) or indeed in a physical phase change (from water into ice or into steam). As opposed to smooth gradual changes, such jumps can be seen as catastrophes, either positive (beneficial) or negative (disastrous).

The simplest Catastrophe is the Fold, which is a simple cubic curve with one maximum and one minimum (Fig. 4), but which can also be laid on its side as in Fig. 5). In Fig. 5), a point moving to the right along the upper branch will eventually jump to the lower branch at point A, because the folded-under part from A to B represents unstable states, while a point moving to the left along the lower branch will continue all the way to point B and then jump up to the upper branch. (Both jumps are indicated by dotted lines.)

Figure 4. Figure 5. Figure 6.

This type of behaviour is typical of a hysteresis curve, in which an approach from one side follows a different path than an approach from the other side: e.g. in magnetization, the transition from paramagnetic to ferromagnetic occurs at a different temperature than the reverse transition from ferromagnetic to paramagnetic. (The ice-water phase transition occurs normally at the same temperature, i.e. the freezing point is the same as the melting point, but it is possible to super-cool water below the freezing point, and with other compounds, such non-equilibrium behaviour is more common.)

Hysteresis means that the system behaves as if there was an inertia, a lag or delay that makes the system want to persist in its status quo longer than it “should”, and than it would if all changes proceeded in a reversible manner. Then, when the system finally “realizes” that it’s walking on thin air and has no business being in that state any more, it abruptly readjusts by jumping to the new state; like a dictatorship, long frozen by terror, undergoing a sudden revolution, while if (in a democracy) slow and orderly change could happen without resistance, no revolution would be necessary. Note that energy is wasted (dissipated) in hysteresis loops because of irreversibility, which makes entropy increase faster.

Now compare the logistic curve (Fig. 1) to the Fold (inverted) (Fig. 6). We can see how the Fold can be generated from the logistic by a simple deformation – by pulling the point on the logistic at which the steep rise ends and the upper plateau begins, sharply to the left, so that the curve folds back on itself. (Deformations are permitted in topology, though not in graphs.)

The equation of the Fold is a cubic, as was already mentioned:

y = x – ax (3)

Comparing equations (1) and (3) (right-hand expressions only), we see that (1) is a quadratic (highest power term is x ) while (3) is a cubic (highest power term is x ); both also have a linear term (x), but no separate constant term, and (3) has no x term.

Is this another way for our Utopian logistic to be converted to a catastrophe? (By a sharp pull to the LEFT?? But the picture is symmetrical; by drawing it the other way, it could be a sharp pull to the right. What matters, probably, is the sharp pull.) Perhaps, but remember that “catastrophe” only means discontinuity. Still, poetic license can be allowed some free range.

The next, somewhat more complex of Thom’s catastrophes is the Cusp, which is the one usually discussed. It is a surface curved in 3 dimensions, and it also demonstrates hysteresis. But there are now two independent variables that vary continuously, while the third (dependent) variable shows discontinuities (jumps). The equation of a Cusp is a quartic, i.e. it has a x term.

y = x – ax – bx ………………..(4)

Its form is like adding the quartic term to the logistic – but this transforms it completely.

I will refrain from discussing the other catastrophes, which cannot be represented at all in 3-dimensional space. Suffice it to say that the Swallowtail has a x term and the Butterfly has a x term. (See my essay on The Swallowtail and the Butterfly for some interpretations.) The remaining 3 catastrophes contain both x and y power terms, but the power never exceeds 4 and for most terms is 3 or less.

The way we think and talk and express ourselves in both the arts and the sciences is profoundly affected by the spirit of the times. So we now have theories of Chaos and of Catastrophe (and even an Apocalypse equation which I did not discuss here), even though in normal times we would only call them theories of non-linearity and discontinuity.

But there is also hope in the midst of despair, as when spontaneous order emerges from chaos – even the striking beauty of the Mandelbrot set. And in the catastrophe series, while the ominous Cusp is followed by the sharpened conflict of the Swallowtail with its “splitting factor” (as we go from x to x ), still, if we persist in continuing on to x in the Butterfly, we arrive at a configuration or Gestalt which, in its unimaginable 6 dimensions, contains a “pocket of stability”; this can be our refuge in adversity, the island of calm in the stormy sea.