Most batch chemical reactions go to equilibrium, either fast or slowly. Most continuous (flow-through) chemical reactions go to a steady state, either fast or slowly. Both equilibrium and steady state can be considered to be a simple single attractor, as the reaction kinetics goes through consecutive iterations of molecular collisions, considered as the consecutive steps or “generations” in the state of the system.

The Belousov-Zhabotinski (B-Z) reaction oscillates between two attractors, made visible as the colours red and blue, with a periodicity of about 1 minute. This is a bifurcation, similar to what we get in iterations of the logistic equation first proposed in population dynamics:

y = rx(1-x)

As r increases, this equation has a single attractor when iterated at low values of r; then bifurcates into two attractors at about 3.24, then shortly after there are 4 attractors, then 8, 16, 32…at shorter and shorter intervals, until at about r = 3.57 the behaviour goes from periodic to chaotic, i.e. the attractor becomes a “strange attractor”. The B-Z reaction represents only the first bifurcation into two attractors, also called a 2-cycle. I don’t know of any chemical reactions that exhibit the next bifurcation (a 4-cycle), or the ones after that.

At equilibrium, classical thermodynamics prevails. If equilibrium is attained in an isolated system, entropy increases during the process of approaching equilibrium, and thereafter remains constant. If we use a continuous chemical process at a relatively slow rate of throughput, the system approaches a steady state. The concentrations of reactants and products remain constant, but the reaction is proceeding at a constant rate, while reactants flow in and products flow out at a constant rate. It is like a bathtub with tap and drain both open, in such a way that the water level remains constant.

Classical thermodynamics does not apply to this system, because the reaction has not come to a stop. This is not an isolated system, it is an open system with respect to flows of both matter and energy. However, while classical thermodynamics does not apply, Onsager’s first-order approximation to thermodynamics does apply. Onsager’s rules continue to apply up to the first bifurcation, as the rate of throughput continues to rise; but these rules are no longer applicable after the first bifurcation, i.e. they do not apply to the B-Z reaction.

After bifurcations begin, we leave the Onsager regime and enter the Prigogine regime of dissipative structures. The system is now considered to be “far from equilibrium”; it has crossed a boundary into a new realm. Although this has not been done in chemical reactions in the laboratory, to my knowledge, further bifurcations t04, 8, 16 etc. attractors (“colours”, if you like, in analogy to the B-Z reaction) would follow with increasing rapidity, until the system would cross into chaos when the number of attractors become infinite, the “colours” kaleidoscopic. (However, see the discussion of “the devil’s staircase” below.)

One could ask, how can one reach infinity in finite time? The answer is, as the steps of bifurcations become increasingly small, they approach zero, and zero times infinity. can be a finite number. The situation is analogous to the Djinns (in Hofstadter’s book “Goedel, Escher, Bach”) delivering the message through an infinite series of Djinns in finite time, because it takes near-zero time to deliver the later messages.

Life exists on the edge of chaos, it has been said. In terms of our picture, this means at a very high number of Prigogine bifurcations, but not at infinity. The “edge of chaos” hypothesis comes from experiments with a sand-pile, which exhibits “self-organization” as more grains of sand are added at the top. Through many micro- and macro-avalanches, the sand-pile maintains a critical shape on the edge of chaos. That shape behaves as a sort of a “meta-attractor”. Similarly, self-organized life maintains. itself on the edge of chaos without sliding either into chaos or into excessively rigidifying order.

In Nina Hall’s book, “Exploring Chaos”, on pages 114-115, the oscillations of colours in the B-Z reaction are considered in more detail. As the flow rate increases, we get complications in the wave form which is more complex than a mere bifurcation into 4 attractors, but at the beginning resembles it. If we identify a “repeating unit” in the wave pattern (somewhat like the unit cell in a crystal lattice, but in time rather than in space), and call the number of peaks in it the “firing number”, we can plot the firing number against the increasing flow rate and obtain a series of steps, a so-called “devil’s staircase”. Its steps come more and more frequently and crowd together, to finally reach, asymptotically, a gently waving almost smooth line at infinity. But the devil’s staircase is a fractal object, for between its treads are further smaller devil’s staircases, and so on at infinitum. I am reminded of my essay Rise and Run about developmental stages featuring crises and plateaus. But here the pattern is much more subtle and detailed, showing it pervading the entire scheme of changes down to the very tiniest. And the crises (rises) are “infinitely rough” (fractally), while the plateaus are smooth, which makes sense. (“Rises” become “Crises” just by adding the initial “C”.)

Again in Nina Hall’s book, on pages 38-39, the first bifurcation of the logistic equation is seen as occurring at r = 3.236…, which is 1 + the square root of 5. It is not clear whether this is the first onset of oscillation or its stabilization, but the author states that the choice of this value of r “was not a random choice” on her part. Now 3.236… happens to be 2 × 1.618…, which is the ratio of two successive members of the exact Fibonacci series which occurs widely in nature, from the shell of the Nautilus to petal whorls in flowers to the spiralling of leaves around stems. Also, 0.618… is the Golden Section, much used in artistic design as an esthetic standard of beauty and harmony: the division of a line of unit length into two parts, a and (1-a), such that 1:a = a:(1-a). Why this irrational number, which has been called phi, recurs in these seemingly very different situations, is not at all clear. Norman Alcock sees the Fibonacci series as the basic pattern of both natural and human development – probably all the way up to Teilhard’s Omega Point. (See the essay “Irrationality” in Section V.)

Now let us go back to certain other connections. In a previous essay called The Key, (Section X) the relationship between exponential growth, exponential decay, sine/cosine oscillations, and damped oscillations is explained in terms of a single simple second-order differential equation which models a restorative force:

d2y/dx2 = ky.

When the restorative force coefficient k is small, the solution is an exponential in x, positive or negative, indicating either self-accelerating growth (a runaway “snowball” or explosion) or an asymptotic approach to the base-line, as in radioactive decay. In the latter case, the base-line is a simple single attractor, while the former case is a “blue-sky” runaway to infinity, which might mean a catastrophe (in the common- sense meaning of a disaster) in the real world. If k is negative and large, the solution to the “Key equation” (my name) is sin x or cos x, indicating an oscillation. The base-line is still an “attractor” of sorts, but the dynamic never reaches it because it overshoots, tries to come back, and overshoots again…and again. Exponential and sine/cosine functions are deeply related not only by the Key equation, but also by the Euler relationship

eix=cosx+isinx

where e the base of natural logarithms and i is the imaginary unit, the square root of minus 1. (The Euler relationship comes from the infinite series for exponentials, sines and cosines, derived from the very basic Taylor series of differential calculus.)

In terms of our discussion here, exponential decay is similar to the single attractor seen at low values of r, near equilibrium, in the regime of Onsager thermodynamics; while the sine/cosine oscillations (undamped) correspond to the state of affairs after the first bifurcation where there are two attractors, corresponding to the crest and the trough of the wave (beginning of the Prigogine regime). However, I don’t think that the Key equation demonstrates any further bifurcations to 4 etc.

The term “bifurcation” occurs in both the Feigenbaum scheme of the successive doubling period of the logistic equation on its way to chaos, and in the Prigogine scheme of dissipative structures evolving to a higher stage through an accumulation of fluctuations, i.e. a “breakdown or break-through” crisis. I think that this is the same meaning of bifurcation in a different but related context. However, I am not sure if the meaning of bifurcation in the sense of Thom’s catastrophe theory is the same, although a diagram on page 155 of the book by Nina Hall indicates that it might be. Certainly the hysteresis observed in Thom’s cusp catastrophe is similar to the overshoot in an oscillation. (Incidentally, “catastrophe” in Thom’s sense is only a discontinuity, not necessarily a disaster. It could be either good or bad, breakthrough or breakdown, opportunity or danger.)

Bifurcation also seems related to the symmetry-breaking which is observed in some physical phenomena, e.g. the separation of the four forces of nature in the very early universe after the Big Bang (this is not a periodicity of 4 attractors; rather, the forces split off one at a time, into 2, then 3, then 4), or the prevalence of matter over anti-matter, or the chirality of dextro and levo enantiomers in living organisms, or the non-conservation of parity in particle physics. Sometimes both branches of the fork are preserved and continue, as in the four forces; sometimes only one persists and the other (almost) vanishes, like matter-antimatter, or the competition between VHS and Beta VCRs. (See the essay “Symmetry- Breaking” in Section X.)

Universality in the Feigenbaum sense means that the period-doubling pattern, and even its scaling constant, is independent of the form of the equation. The graph form does not have to be a parabola as in the logistic equation; just any curve with one hump. (perhaps even half a sine wave?) It seems that iteration bears the same relationship to other arithmetical operation as topology bears to geometry.

We have brought in B-Z clock reactions, period-doubling in the logistic, bifurcation, the Onsager and Prigogine regime, the self-organizing sand-pile, the devil’s staircase, the Fibonacci series, the “Key” equation, Thorn’s catastrophe theory, symmetry-breaking, and scaling universality, in an attempt to bring together the ideas on the place of life in the scheme of things. We could have also brought in feedback (positive and negative), autocatalysis, cycles, homeostasis, and emergence, to mention a few more. Where did all this “putting it all together” get us?

How close to the boundaries of chaos dare we go before falling in? In one sense, we are very close in absolute distance, being in the thickening “crust” at the boundary where the successive bifurcations are compressed; in another sense we are an infinite number of tiny steps away. Very close and yet very far. And I have also seen it mentioned, in Nina Hall’s book, that the boundary between order and chaos is itself very complex and fractal – in fact that it is represented by the entire Mandelbrot set.

We are between equilibrium and chaos, climbing the devil’s fractal staircase – to the omega point, or to dissolution?