A Brief History of the Concept of Chaos

Huajie Liu  (huajie@phil.pku.edu.cn)

(Department of Philosophy, Peking University, 100871, Beijing, P.R.China)

In the past two decades, with the publication of more than 7200 papers and 260 books, the discovery of chaos in nonlinear dynamics has made an overwhelming impact on many disciplines, including mathematics, mechanics, computer science, biology, ecology, astronomy, engineering, economics, art, and of course philosophy. Only the shock wave of quantum mechanics in the history of sciences can compare favorably with it in the sense of making a great stir in society and media. Like quantum mechanics, chaos theory caused a lot of semantic confusions when it became a hot and popular topic.

In this paper, I want to clarify the meanings of chaos through materials from ancient cultures to modern sciences and outline the nature and significance of it. First, it should be pointed out that chaos is nothing else but a kind of special dynamical motion coming from deterministic equations, displaying extremely sensitive dependence on the initial conditions of the system. There are many considerations about the concept of chaos, but this is its core meaning from the point of view of present science and mathematics.

The reader, who is not concerned with the semantics of chaos in ancient times and general sciences, may skip the first two sections, and can directly go to section 6, 7 and 8, in which I talk about the definition of chaos and its philosophical significance.

1.Chaos in Some Ancient Classics

Hesiod on the standard lines 116, 123, 700 of Theogony said Chaos is the primitive god before Gaia, Tartaros and Eros, and Publius Ovidius Naso at the beginning of Metamorphosis repeated the idea.

The Holy Bible mentions the conflicts between God and chaos on many occasions e.g. in Genesis 1:1-2; 26-30, Isaiah 24:9-10,16-17; 34:9-11, 45:18-19, Jeremiah 4:23-26, Micah 1:5-6, Zechariah 14:3-5, Job 10:20-22 and John 1:5-10. The authors of the Bible describe chaos as "a place of disorder", "without any order", "the state of formless", "of utter disorder and confusion", "the condition of emptiness, unreality, and desolation", and "a meaningless existence".

The Bible doesn't directly tell us who created chaos. Maybe it already existed before the creation and it would exist forever just in order to resist God's will. So there are two speculations: one is that God began his creation from a primitive base that was an undifferentiated mixture, i.e. chaos; another is that at the beginning there was only the soul of God, the heavens, and that earth and chaos were all generated from God.

I find that nearly all nations in the former times constructed their own fairy tales about chaos creation with similar structures. Here I want to tell a chaos fable excerpted from one of the ancient Chinese classics Chuang-Tzu:（《庄子》）

In an older ancient document Mountain-Sea Sutra （山海经）（ Shanhai Jing) chaos indicates the God of Sun:

2.The Use of Chaos in the General Sciences

To my knowledge, many scientists including some social scientists used the concept of chaos in very different ways.

Aureolus Paracelsus used chaos referring to air and J.B. van Helmunt created the concept of gas from chaos. Thomas Burnet, Robert Boyle, C.Linnaeus, I.Kant, K.Marx, T.R.Malthus, John Stuart Mill all used chaos to describe their theories or philosophical beliefs. Here I only cite Mill's sentences from chapter 7 of his Logic:

In 1938 and 1943, Norbert Wiener wrote two mathematical papers The Homogeneous Chaos and The Discrete Chaos in Amer. J. Math. , in which he coined the words and phrases "chaoses", "homogeneous chaos", "one-dimensional chaos", "multidimensional chaos", "pure chaos", "discrete chaos", "polynomial chaos" etc. and set up his theory of chaos stochastic process, modeling a new paradigm contrary to Newton's deterministic mechanics tradition.

As the theory of dissipative structures became fashionable from the 1970s, Ilya Progogine's concept of entropy chaos propagated all over the world. Before the 1980s, Prigogine often mentioned chaos in his two famous books with two distinct meanings: 1) the thermodynamic equilibrium chaos, and 2) non-equilibrium turbulent chaos. I think he is partly responsible for the arbitrary and careless usage of chaos in academic fields.

3.Before the Current Wave of Chaos

It is absolutely unwise to declare who was the first to recognize nonlinear dynamic chaos in the history of sciences even though some people prefer to do so, e.g. T.Y.Li and James Yorke were considered to be the first people discovering chaos in 1975. In fact before Li & Yorke, Robert M.May, a theoretical physicist from Australia and then a ecologist in America, published a paper "Biological populations with non-overlapping generations: stable point, stable cycles and chaos" in Science, in which he suggested that chaos was a new steady state of dynamical system, a key idea for the development of this enterprise.

There was an interesting wide-spread story about Yorke & Li and R.May: the publication of Li & Yorke's paper "Period Three Implies Chaos" was encouraged by May's address at Maryland University in 1974.

These are relatively recent happenings. From the perspective of history and philosophy of sciences, we must focus on many facts and materials laid out before us by some great scientists, such as J.C.Maxwell, A.M.Liapunov, P.M.M.Duhem, J.-S.Hadamard, Henri Poincaré. It seems to us even astonishing how they could possess so precise an understanding about the instability of classic mechanics long before ordinary scientists in the 1970s.

Maxwell argued that it is a metaphysical dogma that we can get the same consequent from the same antecedent. No one can deny it, but actually the same antecedent never emerges twice. In physics we use a similar postulate: we can get a similar effect from a similar premise. But this time we change from sameness to similarity, from absolute accuracy to rough approximation. This transformation, does not cause any trouble, but there are other cases in which we will face complex instability.  James Clark Maxwell (1873) said:

It is a metaphysical doctrine that from the same antecedents follow the same consequents. No one can deny this. But it is not much use in a world like this, in which the same antecedents never again concur, and nothing ever happens twice.... The physical axiom which has a somewhat similar aspect is "that from like antecedents follow like consequents". But here we have passed from sameness to likeness, from absolute accuracy to a more less rough approximation. There are certain classes of phenomena, in which a small error in the data only introduces a small error in the result, the course of events in these cases is stable. There are other classes of phenomena which are more complicated, and in which cases instability may occur, the number of such cases increasing, in an extremely rapid manner, as the number of variables increases.... Every existence above a certain rank has its singular points: the higher the rank, the more of them. At these points, influences whose physical magnitude is too small to be taken account of by a finite being, may produce results of the highest importance. If, therefore, those cultivators of physical science from whom the intelligent public deduce their conception of the physicists are led in pursuit of the arcana of science to the study of the singularities and stabilities of things, the promotion of national knowledge may tend to remove that prejudice in favor of determinism which seems to arise from assuming that the physical science of the future is a mere magnified image of that of the past. (From Teaching Nonlinear Phenomena- I, pp.47)

Maxwell suggested that there exist unstable systems whose behavior may conflict with the general causality postulate in physics, which I will discuss later in the context of significance of chaos.

In 1898 a French mathematician Jacques Hadamard at the age of 30 found a negative curve system displaying sensitive dependence on the initial conditions. At the same time a scientist, historian and philosopher of science Pierre Duhem grasped the true philosophical meaning of Hadamard's result. This led him to distinguish physical point and trajectory from mathematical geometrical ones.
 

Now we must turn to talking about the father of modern nonlinear dynamics, Henri Poincaré, who contributed lots of tools and methods to the exploration into a new world of classic mechanics and mathematics. Poincaré created qualitative dynamics, ergodic theory, topology and bifurcation theory, all of which are fundamental to the study of the current chaos theory. He not only described how the dynamical instability influences prediction in his famous scientific philosophy works but also specified homoclinic transversal lattice structures in his Les Méthodes de la Méchanique Céleste. Henri Poincaré said in his 1903 book Science and Method that:
 

In 1920s van der Pol and van der Mark in the studies of relaxation oscillations found phenomenon of frequency demultiplication. They published a very short paper in Nature in 1927 whose importance lies in that 1) they found period-doubling bifurcation in a physical system that is considered a basic scenario to chaos, and 2) they draw out the so-called devil-stairs image with fractal structure.
 

English mathematicians M.L.Cartwright & J.E.Littlewood (CL) absorbed Pol & Mark, Morse, Birkhoff and Morman Levinson's results and found the "bad curve" in researching into nonlinear differential equations of second order. N.Levinson (L) expanded CL's research in his 1949 paper "A second order differential equation with singular solutions" in Annals of Mathematics in which he proved successfully that there exists an attracting chaotic solution with continuum power that was called strange attractor after 1970s. S.Smale abstracted an important concept "horseshoe" in 1960s from CLL's work.
 

4.Two Clues to the Fad of Chaos Theory
 
 

The immediate predecessors of the washing craze for the gold of chaos theory starting in the 1970s came from two different kinds of developments at the beginning of 1960s. One is the KAM theorem of conservative systems, and the other is E.Lorenz's weather model of dissipative systems.
 

KAM theory dates back to A.N.Kolmogorov's address in the International Mathematicians Congress at Amsterdam in 1954. V.I.Arnold (Kolmogorov's student) in 1961 and J.Moser in 1963 proved this theorem rigorously. Hence the name KAM. The KAM theorem was too difficult to understand even for professional mathematicians. Scientists did not grasp the spirit and exact meaning of it until about a decade later. KAM theorem, or more accurately the violation of it, can be used to explain many stochastic behaviors in classic conservative mechanical systems. In 1969 G.H.Walker and Joe Ford exploited KAM theory to interpret some results from computer experiments. In their paper "Amplitude instability and ergodic behavior for conservative nonlinear oscillator systems" the authors established contacts among the KAM theorem, the Poincaré recurrent theorem, the foundation of statistical physics, the origin of complexity in simple nonlinear system, M.Hénon & C.Heiles experiment and FPU phenomena etc.
 

In the approach of dissipative system, E.N.Lorenz in 1963 wrote a paper "Deterministic nonperiodic flow" in J. Atmos. Sci. in which he gave a simple ordinary equation system displaying a butterfly-like strange attractor (David Ruelle and F.Takens coined the term in 1971) that has become the icon and symbol of chaos theory. About at the same time a Japanese graduate Yoshisuke Ueda (上田）independently found a strange attractor in the Duffing equation.
 

The advancements of the above two roads all have been drawing supports from numeric simulation. It is with the aids of calculating and displaying technologies of modern computer that chaos theory and other nonlinear science theories grow up.
 

5.Logistic Map: Nonlinear Sparrow
 

As a Chinese proverb goes: "A sparrow may be small but it has all the vital organs ---- small but complete." The one-dimensional logistic map is just the sparrow of nonlinear sciences, which following the two-body model and Brownian motion, can be regarded as the paradigm of the third breakthrough of classic mechanics. Logistic map possess bifurcations, stable and unstable periodic orbits, periodic windows, ergodic and mixing behaviors, homoclinic connections, chaotic orbits and some kinds of universality.
 

G.Julia, von Neumann, S.M.Ulam studied logistic map long before the rise of chaos theory. In 1973 three mathematicians MSS (N.Metropolis, M.L.Stein and P.R.Stein) in Los Alamos undertook researches on transformation on the unit interval. They introduced systematically symbolic dynamics, constructing harmonic and anti-harmonic algorithms for symbol sequences. After MSS, the DGP's (B.Derrida, A.Gerois and Y.Pomeau) paper published in 1978 was an important advance, which gave an internal self-similar theorem that indicates that there exists a kind of topological universality in nonlinear dynamics. M. Feigenbaum in the midst of 1970s found constants d and a in logistic map that represents another kind of universality, metric universality, in nonlinear systems.
 

At that time Li & Yorke's article "Period three implies chaos" caused a little shock in the academic circles and was copied everywhere. In fact, A.N.Sarkovskii, a Ukrainian mathematician, published in a not famous journal a more general result than that published by Li & Yorke a decade years ago. Around logistic map there are many misunderstandings. Here I want to clear up some of them.
 

(1) The Li & Yorke theorem says the existence problem of periodic orbits given a proper parameter. It does not directly deal with stable orbits' distribution from different parameters, so it does not correspond to the bifurcation figure that can be easily seen and displayed on computer screen. Only on the product space of parameter space and phase space can we look up the bifurcation spectrum. Li & Yorke theorem only talked about the existential property of periodic orbits on phrase space.
 

(2) The Li & Yorke theorem and the Sarkovskii theorem discussed the existence of periodic orbits but they asserted nothing about the situation of their stability and did not say their measures related to chaotic motions, i.e. what percent each one accounts for. For one dimension unimodal map satisfying H.A.Schwartz conditions D.Singer in 1978 proved there is at most one stable orbit given a parameter. Therefore there exists this possibility: the system may possess periodic orbits, but many or even all of them are invisible because only stable orbits can be seen physically.
 

(3) The Li & Yorke theorem talked about the vertical property of bifurcation spectrum of logistic map and Sarkovskii theorem talked about the horizontal property of it. Some persons think Li & Yorke theorem means that stable periodic orbit coexists with chaotic orbit at a fixed parameter. This is wrong. For logistic map for a parameter within period three window there is no stable chaotic orbit and any other stable periodic orbits.
 

(4) The Li & Yorke did not define chaos as a serious scientific term. Chaos is still a routine concept even in this very key paper for the great fuss of chaos theory.
 

6. What Is Chaos?
 

There are many working definitions for chaos. In the framework of classic mechanics, chaos often means deterministic chaos. Four quantitative properties can be used to identify whether we face a chaos system. The four indexes are positive Liapunov characteristic exponents that measure the separation speed of nearby trajectories, fractal structure and dimension of the phrase space, not negative topological entropy, and continuous power spectrum. I also induce five general characteristics for chaos motion:
 

(1) Determinism
 

Theoretically, the chaos motion studies in classic framework must be generated by one or more deterministic equations that do not contain any random factors. The system states of past, present and future are controlled by deterministic rules. It is because scientists find stochastic-like behavior in completely deterministic system that they become excited and are attracted to probe into the secrets. In the long run, scientists must study more complex systems that include the type now called chaos. But at present it is of utmost importance to strengthen determinism.
 

(2) Nonlinearity
 

Nonlinearity means the non-superposition of factors or effects; it means there are terms like
 

X ², 2mxy, byz
 

in the equations, in which x, y and z are variables, b and m are parameters. Nonlinearity is a necessary, but not sufficient condition for the appearance of chaos. Chaos motions must come from a nonlinear system but nonlinearity does not necessarily imply chaos. Piecewise nonlinearity is not equal to linearity.
 

(3) Sensitive Dependence on Initial Conditions
 

Generally, the evolution of a system depends on its initial state even though some can demonstrate equal final outcomes. When our interest focuses on the whole dissipative attractor the evolution seems not sensitive dependent on its initial states because all trajectories fall onto the attractor. But if our interest focuses on the interior structure of the strange attractor we can get a peculiar picture: trajectories diverge and converge exponentially. For a chaotic system, this property of SDIC must be valid for nearly almost all-possible initial states. Under this constraint a possible geometrical explanation for chaotic phase structures is stretching and folding. The chaotic trajectories move within the finite phase space forever. O.E.Rössler thought that perichoresis (around-motion) used by ancient Greek philosopher Anaxagoras is appropriate for this kind of motion and if G.D.Birkhoff had known this usage he would have not created a Latino term recurrence.
 

(4) Aperiodicity
 

Chaotic motion is a new topological type of motion that is very different from fixed point, limit cycle and limit torus. Its orbits are non-periodic. This means that a chaos orbit can never join another one or repeat its history. But not all non-periodic orbits are chaotic orbits. Almost periodic motions and quasi-periodic motions are aperiodic but not chaotic. Here I want to clarify one point: that the chaos system may (and often does) contain periodic motions, and periodic and non-periodic orbits can twine together.
 

The above four properties are necessary but not sufficient for chaos.
 

(5) Some Stability with Some Tension and Boundness
 

There are different attitudes towards the requirements of the stability of chaos. From the perspective of pure mathematics, without consideration of stability of orbits is convenient to definite chaos. But physically, stability is so important that it is better to include stable constraints in defining chaos motions. In fact scientists tend to understand the chaos motion physically. In their eyes, chaos is bound, involving a kind of loose stability. The chaos motion on the strange attractor is locally unstable but globally stable.
 

R.L.Devaney's definition of chaos in 1986 includes three parts: unpredictability, indivisibility and regularity that correspond to SDIC, ergodicity or mixing of phase space, and deterministic rules of system respectively. Lastly, I will try to give a descriptive definition of the classic chaos:
 

The chaos motion is a recurrent, random-like, and aperiodic behavior generated from deterministic nonlinear equation with sensitive dependence on initial conditions of system. From viewpoint of algorithm information theory chaotic trajectory possesses weak algorithm stochasticity.
 

7. The Destroyer of Operational Causality
 

David Hume (1711-1776) discussed the problem of cause and effect in his book An Inquiry Concerning Human Understanding. He said the knowledge of the relation of cause and effect is not attained by reasoning a priori, but arises entirely from experience, when we find that any particular objects are constantly conjoined with each other. 

The law of causality is the premise for all scientific researches but no empirical facts can falsify it. It is always logically correct. In my opinion, the law of causality means that the same cause can produce the same effect. The problem is how to determine what is the same cause and what is the same effect. Rigorously speaking, the same spatio-temporal conditions and other physical situations can not take place repeatedly in the real world. Then what does causality mean? It must mean that a very similar cause should produce a very similar effect in the sense of operation. However, what is "very similar" and how can we guarantee the realization of this kind of very similar? In other words, have we enough reasons to accept the law of operational causality?
 

Before the chaos theory came into being, this question was not even asked seriously in the scientific community. Philosophy of quantum mechanics met with this problem in a different context. Now the chaos motion because of its sensitive dependence on its initial conditions definitely violates the operational law of causality. In the chaos system, a small derivation produces a great derivation. In a Chinese idiom it is said that "an error the breadth a single hair can lead you a thousand mile astray".
 

Moreover, does chaos motion violate the law of causality itself? I don't think so. At least the present chaos theory does not support this bold assertion.
 

If the chaos motion does destroy the law of operational causality, it will impose a vital limitation on all senses of prediction theories in sciences and engineering practices. If the system under consideration satisfies DMB condition all are OK. But if it does not satisfy DMB condition we must re-examine the conclusions people reached before. DMB represents P.M.M.Duhem, J.C.Maxwell and L.Brillouin respectively who pointed out the very meaning of dynamical instability in sciences very early. DMB condition (I coined this phrase in 1994) represents condition that the system does not possess the unstable dynamical process.
 

8.Deterministm and Predictability
 

Many philosophers of science have cited Pierre-Simon Laplace's famous description of determinism in Essai philosophique sur les probabilités and made a lot of strange remarks. I feel that Laplace's words are represented by the sentence structure "if something then something" with the subjunctive mood that resembles the hypothesis of causality. He did advance the deterministic paradigm to the supreme stage but his sentence can not be falsified directly. In fact it is only a belief and can never be falsified.
 

After all sciences are dealing with prediction affairs. Is it because of chaos motion that it is impossible to predict the future state of any system? Absolutely not! There are many kinds of predictions. Chaos only imposes some limitation on some prediction. Even in the sense of trajectory prediction, it is not sentenced to death by chaos theory yet. A finite tracing to chaotic orbit locally is always possible. I derived a formula in 1994 for chaos system:
 

T = M / ( l - L )
 

in which T is critical time, ( l and L are A.M.Liapunov exponent and R.O.S.Lipschitz constant respectively, and M is also a constant. If prediction time for a chaotic system goes beyond the critical time T of the system then this prediction in the sense of trajectory is invalid. If prediction time is less than the critical time T then the prediction is possible or maybe acceptable.
 

9. Chaos and Metaphor
 

From the 1980s chaos models began to replace the role of period models in natural sciences, economics and social sciences because chaos models logically contain fixed point (equilibrium), limit cycle (period) models and at the same time they can also simulate more complicated process.
 

Along with other parts of nonlinear sciences chaos theory has more or less changed the scientific methodology. Chaos theory uses reduction methods but it transcends any reductionist dogma. In the whole area of chaos studies, the relationship between global behavior and local behavior, reduction and emergence, order and chaos, predictability and unpredictability is discussed theoretically and practically in a natural way.
 

Chaos studies also give impetus to change in values. In natural and artificial systems, which motion is better? Is the equilibrium point, stable period or chaotic motion? Over a long period of time, equilibrium and periodicity has been considered as good states. Now we can find that in some cases unstable, chaotic motions may be more useful and valuable.
 

Not surprisingly, the outlooks of chaos theory and general ecology are regarded as a strong support to the theory and practice of sustainable development in academic fields. Some postmodernists and feminists and mysticism borrow nearly arbitrarily fashionable scientific concepts, including chaos, nonlinear, bifurcation, fractal, to argue for their beliefs. There are many misunderstandings surrounding the chaos theory in its translation from strict mathematics and natural sciences into cultural criticism.
 

Undoubtedly, some philosophers, literary or art critics and mystics use the chaos theory as a metaphor. Nobody can forbid the use of scientific terms and metaphors. In this situation, we should take a tolerant and skeptical view of some claims.
 

This paper does not touch upon debates on quantum chaos. I think classic chaos and quantum chaos (if there ever exists such a chaos) fall under two different descriptive categories of sciences. Classic mechanics is deterministic from head to heel and only the initial conditions allow for the possibility of randomness. However, quantum mechanics is probable in its foundation. Probability as a basic premise is introduced into the quantum theory, but the formal system of quantum mechanics itself is also deterministic. Especially the evolution of quantum probability flow is deterministic. No matter where we may go, the debates on quantum chaos have enabled us reconsider the validity of the correspondence principle. We feel that now man is far from finding the true story of the nature.
 

10.Brief Conclusions 

The concept of chaos in our cultures has very rich implications, but in modern chaos theory of nonlinear dynamics chaos is a clear term with definite meanings. In my opinion chaos is a kind of simple behavior compared to other many unknown processes in physical systems. It is still a difficult question whether there is any new type of steady state between more complicated movements and present chaos movement. Mathematicians will answer this question in the future. 

Chaos theory has nothing to do with so called postmodernist theory. The most important philosophical impact of chaos on scientific reasoning is its violation of operational causality. It confirms David Hume's worry.

Main References

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