Discrete Dynamics in Nature and Society, Vol. l, pp. 9-15Reprints available directly from the publisherPhotocopying permitted by license only

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Chaos: Challenges From and ToSocio-Spatial Form and Policy

DIMITRIOS S. DENDRINOS

Urban and Transportation Dynamics Laboratory, 107 Marvin Hall, The University of Kansas, Lawrence, KS 66045-2250, USA

(Received 9 October 1996)

A brief assessment is given of the major accomplishments made through the mathematicsof chaos to the understanding of socio-spatial dynamics to date. Certain shortfalls arealso presented, mostly associated with model testing and falsifiability which transcendsocio-spatial dynamics. Beyond such shortcomings, lie an array of challenges for chaoticdynamics involving specifically socio-spatial form and policy. A few directions on meet-ing these challenges are suggested including the case of limited chaos.

Keywords: Chaos, Socio-spatial dynamics, Complexity, Entropy, Limited chaos

1 INTRODUCTION

This is the inaugural issue of a journal set up tocarry out a formidable and unique experiment: tosearch into the possibility of integrating the socio-spatial and natural sciences through the (univer-sal?) method of discrete chaotic dynamics or,more broadly, nonlinear dynamics. No doubt,such an attempt will be met by considerable chal-lenges. By and large, the first challenge will be todemonstrate the validity of such a method forsocio-spatial analysis beyond mere speculativetheoretical hypotheses, by no means an easy task.

Socio-spatial dynamics lack a Belousov-Zhabotinskii chemical reaction, which so clearlydemonstrates the emergence of pattern out ofwell understood nonlinear interactive dynamicsoff equilibrium states with such an impressive

and eloquent manner. No matter how much so-cio-spatial analysts yearn for and seek equiva-lently demonstrative examples (by looking forinstance into stock market time series or urbantraffic flow patterns), socio-spatial evidence carry-ing such convincing power may never comeabout. Even if one argues that socio-spatial ana-lysts are looking in all the wrong places, therecan be very little doubt that if any good places ortimes to look for such patterns exist, there mustbe very rare if any at all: social reality simply isnot that neat. This is the null hypothesis.Those of us in the social sciences in general,

and the human spatial sciences in specific, whochoose to use nonlinear dynamics in modelingand interpreting socio-spatial events in a mannerwhich draws from the power and (albeit non-

social) beauty of mathematical chaos have done

10 D.S. DENDRINOS

so thus far merely out of eclectic preferences, andon obviously tenuous grounds. And we will bethe first to admit it. In our speculative journeys,we search for qualitative properties and rigorousmathematical statements about socio-spatial dy-namics utilizing certain key elements of chaos. Itis of interest to note that all of us who approachsocio-spatial dynamics from this perspectivecome from different angles (initial conditions).We do so only because something quite compel-ling attracts us to it. A faint hope and possiblyweak expectation characterizes our quest, thatsomehow we are to acquire innovative insightfrom such an endeavor.

But why are we so much drawn by mathemati-cal chaos? Aesthetic appeal and fashion argu-ments aside, it is because of its promise: we are

betting that when and if such insight does everreveal itself and is appreciated by a wide enoughaudience, it will have a powerful and profoundimpact upon social action. It will compare onlyto that brought about by Malthus, the end ofwhom this construct indeed foretells. And socialaction is the ultimate motivation of all socialscientists, transcending all social theories, nomatter how well these theories fit the real world.After all, real world phase space and modelworld phase space are not, and they may neverbe topologically equivalent.

2 SOCIO-SPATIAL COMPLEXITY

For long, social sciences have acknowledged theobvious, namely that the key ingredient in thesoup of social systems, as well as in social action

(at both the individual and collective levels) iscomplexity. Far beyond adding flavor to socialsystems and actions, complexity is contained inthem for good reason: it offers them a high de-gree of freedom, and it is as if these systems andactions have an inherent and inalienable right tocomplexity, a multidimensional concept.To the immense micro- and macro-human

heterogeneity, if one adds the vast spatial hetero-

geneity of our planet in its micro- and macro-

geographies, and superimposes all this to thestartling differences in micro- and macro-socio-global time periods- in all what we like to call"social spatio-temporal heterogeneity" then onebegins to appreciate our predicament and questin finding simple patterns out of the complexityof human behavior. Incidentally, this spatio-tem-poral heterogeneity is a strong enough reason tosimulate socio-spatial dynamics in discrete spaceand time, hence the title of this journal.We are forced to seek any help we can get (let

alone inspiration) from all sources, including thenatural sciences and mathematics. Mathematicalmodeling of complex systems seeks to find andisolate simple patterns of behavior to at least for-mulate some hypotheses at some aggregative(coarse) scale, hypotheses which we find of interestfor various reasons and within certain contexts.Our interest in mathematical chaos can be well

founded by drawing on certain very articulatearguments made in an interesting and recentbook by Cohen and Stewart (1994). There, theauthors try to derive some general principles onhow to link simplicity of laws governing the be-havior of physical systems to their underlyingcomplexity. They do so because they wish to gobeyond natural sciences and its reductionistmethods, and derive a general theory of complex-ity which could encompass the social sciences aswell. Simplicity in aggregate behavior of certainsystems can be and has been detected, theauthors argue, while the disaggregative (finebreakdown into) parts of these systems are in-volved into a complex interplay oftentimes farbeyond our full understanding, let alone our cap-abilities to .model.

Certain qualitative properties of chaotic dyna-mics, sensitivity to initial conditions and the pre-sence of chaotic attractors in particular, seem tostimulate suggestive linkages between the simpleupper level laws of nature and the complex lowerlevel interplay of the numerous parts of a systemunder investigation. To Cohen and Stewart theselinkages between complexity and simplicity are of

CHALLENGES IN SOCIO-SPATIAL SCIENCE 11

special interest because they seem to also suggestan alternative (nonreductionist) way for ap-proaching the study of these systems. Nonreduc-tionism is of interest to socio-spatial analysts as

they encounter considerable obstacles in beingpure reductionist of any note.Cohen and Stewart devised two new concepts,

"simplexity" and "complicity" to show how, de-pending on a context, two types of patterns canpossibly emerge in the study of complex systems.The first pattern is drawn from the legacy ofchaos which highlights the extreme dependencyto initial conditions. Simplexity emerges when aninherently (at a lower level of analysis) complexsystem, comprised of a vast number of almostidentical parts (rules), has simple (upper level)laws (one outcome) which describes it well givenour context of choice. The perennial example ofNewton’s law of gravitation is cited, given how itcan (accurately to an acceptable approximation)describe any planet’s orbit, although a planet isan aggregate of a very large number of (almostidentical in their position in space when viewedfrom a long enough distance) individual particles.On the other hand, complicity emerges when aset of many vastly different parts (rules) can gen-erate a simple aggregate outcome. This case high-lights the presence of specific strange attractorsin phase space of dynamical systems under a

chaotic regime.This approach to complexity is clearly one of

the legacies of mathematical chaos. Cohen andStewart refer to complicity as chaos "collapsed"and they speculate that the still elusive Theory ofEverything in physics will contain such a compli-city pattern. They go even further and attributethis mode of thought to the manner in which thehuman mind (or consciousness) really works.Well, a social scientist might not be so sureabout the last claim, but few will doubt the fruit-fulness of a search for a nonreductionist compli-city type model of human societies.

All the efforts recorded so far by socio-spatialanalysts, see the latest work contained in a Spe-cial Issue on "Nonlinear Dynamics in Urban and

Transportation Analysis" in the journal Chaos,Solitons and Fractals (1994), employing conceptsand ideas from chaotic dynamics, point to thisnonreductionist premise as their pure (and possi-bly only) justification. Be that inter- or intra-urban population dynamics, dynamical centralplace theory, dynamics of urban form and struc-ture, inter-regional linkages through trade dy-namics, technological spatio-temporal diffusion,or any other socio-spatial topic attracting currentattention in the literature, all share a commonstarting point: the suggested model form doesnot emerge out of extensive experimentation orthrough analyzing vast data sets, but rather outof theoretical speculation.

3 THE CHALLENGES

There is still no empirical evidence to make thetransition to a reductionist mode, while dealingwith social dynamics under any social science per-spective, no matter the numerous claims to thecontrary from many a source. In that sense, weare forced into seeking rigorous nonreductionisttheories. No matter their theoretical pedigree,some theoretical propositions enjoy a wider re-

cognition simply because social scientists have lib-eral degrees of tolerance for approximations theirmodels and outputs carry with reality, a tolerancewhich in any case is quite higher than that re-

quired for modeling in the natural sciences.To the extent that the social mathematical

chaos literature is concerned, all available statisti-cal evidence presented to date is at least inconclu-sive, see also the book by Kiel and Elliott (1995),where some evidence from economics is assessedwith similar conclusions. What may seem like pat-terns which, if allowed to go on for a long enoughtime period, may demonstrate chaotic behaviorand fractal properties can also be viewed as mereoscillations, indistinguishable from various typesof noise. It is so in stock market data and intraffic flow data, two social time series that are

12 D.S. DENDRINOS

available in relatively frequent counts to evenstart asking testable questions.

In reality there is X6zocr, but it does not possessthe properties (or the oddities as some might say)of mathematical chaos. But this difference maynot be as damaging as it may sound at first and,to the contrary, it might even be quite helpful tobetter understand both real chaos and reality.For instance, what is (in reality) determinism,randomness and uncertainty may be better un-

derstood by looking at the deterministic structureof simple chaos bearing mathematical models.

Efforts to unpeel "layers of determinism" ledKellert (1993) into envisioning "bounds to predict-ability" for complex systems even simply statedones, in presence of apparent order under thechaos demon, and thus rendering our understand-ing of these systems quite limited let alone discre-diting any efforts towards prediction. Chaoticregimes point out dramatically how errors in esti-mating the state variables might be more due tothe errors in the estimation of parameters (primar-ily due to levels of approximation involved in suchparameter estimates aside from their mis-specifica-tion), so much so that they might compromiseerrors in the state variable(s) due to slight shift ininitial conditions (or their approximations). Andall this, in addition to model mis-specification dif-ficulties, which are rendered almost impossibleunder chaotic conditions. In effect, chaos points tothe flaws of the very method of estimation.

4 PROSPECTS

This journal’s tall order, to attempt an integra-tion through discrete nonlinear dynamics of thenatural and socio-spatial sciences by disprovingthe null hypothesis, must however proceed be-cause of the high stakes involved, even if theprospects as of now for its eventual success maybe judged to be limited. The reasons are indeedquite obvious, and of potential interest to naturalscientists as well: if Physics can produce a Theoryof Everything this Grand Theory must link all

components of this Universe at all its scales,from the nanoscale to the cosmological scalethrough the mesoscale of human societies; nomatter how informative such a Theory mightturn out to be it must address our scale too.And if a Theory of Evolution is such a general

theory as its advocates proclaim, then such a

theory must link all scales too, and in particularall these mesoscales neighboring our own scale atthe level of human societies.

So far, and beyond these grand themes, thegeneral method of nonlinear interactive dynamicshas failed to address subjects of immediate andpractical interest to those who directly impactsocio-spatial form, structure and policy or action.The form and architecture of buildings, settle-ments, cities, nations; the structure (economic,social, political or any other one) of individuals,communities or any other social aggregates; theconcrete social actions or policies by individualsand governments, all of the above have remainedaloof from the plows of nonlinear interactive dy-namics and chaos. The challenge of nonlineardynamics is to demonstrate its relevance to con-crete policy, management and design.

Socio-spatial dynamics have remained so fartoo abstract; efforts to produce certain universalrules have neglected (some even would argue thatnonlinear dynamics are incapable of) addressingspecific subjects of social systems because of itsgenerality, its complicity. Maybe, like the expec-ted Theory of Everything, it might illuminate uson nothing. But this criticism could be leveledagainst all mathematical modeling of socio-spatial systems. Be that as it may, socio-spatialdynamics must first demonstrate that they are notmore guilty of such neglect than at least the aver-

age mathematical model of social systems. Andafter having done so then they can march ahead.

5 CERTAIN SUGGESTIONS

Numerous lines of promising research are cur-rently underway, some presented in the 1994 issue

CHALLENGES IN SOCIO-SPATIAL SCIENCE 13

of Chaos, Solitons and Fractals, as already men-tioned. Here, I will confine myself to only twopossible outlets for fruitful extensions: the subjectof entropy in socio-spatial systems, and thesubject of limited chaos in a largely inter-connected network of spatio-temporally interact-ing stocks.How is entropy, of the information type a la

Shannon, behaving in socio-spatial systems andwhat principles of extremum are implied in suchgoverning potentials (if any) is a subject of someinterest to social analysis. For one, it supplies apossible linkage between the natural and socialsciences, as indicated for example by the two Edi-tors of this journal, Sonis and Gontar (1992). Onewould like to see, however, what is the exact inter-pretation of these principles and what do they im-ply for social action under specific contexts.The second subject is that of limited chaos, a

topic which has not attracted much attentionamong either mathematicians or natural scien-tists, let alone social scientists, and first identifiedby Dendrinos and Sonis (1990) in the case of ageneral discrete nonlinear relative dynamics mapcontaining interactions among one stock’ shares

distributed over three locations. More broadly, itis possible to show that in a rather extensivelyinteractive nonlinear dynamical system, somestate variables might experience chaotic behaviorwhile others are in relatively calm dynamics.Some examples of this phenomenon are shown inFigs. 1, 2: in Fig. the case of a "rod" type, andin Fig. 2 the case of a "sheet" type dynamics aregiven. Both cases correspond to limited chaosplotted on the unit cube, and are based on thethree stocks and two locations version of themap, see Dendrinos and Sonis (1990):

xi(t + 1) Fi(t)/[1 + Fi(t)], i= 1,2,3,

where

lnFi(t) lnAi + Z{aijlnxj(t) + b0-1n(1 xj(t) ],

i,j--1,2,3,

O < xi(t) < 1, Fi(t) > O, Ai > O,-oo < (ao.,bo.) < +oo.

In this map, Fi(t) depicts the locational compara-tive advantages (a Darwinian notion) of stock xi

FIGURE Rod-type dynamics of limited chaos; 10,000iterations. See text for parameter specifications.

FIGURE 2 Sheet-type dynamics of limited chaos; 10,000iterations. See text for parameter specifications.

14 D.S. DENDRINOS

at a designated location (one of two competing forthe three stocks locations) at time t. Parameters aand b designate elasticities of interaction (compo-sites of the prevailing interaction forces, a physicalnotion) among stocks and between locations (dueto both size and location). The parameters A iden-tify environmental constants (representing differ-ent socio-spatial universes).Under the parameter specifications:

[a]2 0 10 -2 0

0 0 -2

[b]0 0 10 -1 0

0 -1 -2

A (1.0-4, 10-4, 10-3), x(0) (0.3, 0.1,0.2),

one obtains the "rod" type dynamics of Fig. 1,identifying a two-dimensional (for stocks x2 andx3) set of points where trajectories are attracted(stable periodic motion), and nonattractor boundchaotic behavior for the xl stock.

In Fig. 2, and under the specifications:

[a]0.2 0 10 -2 0

0 0 -1

[b]0 0 10 -1 0

0 -1 -1

A (10-4, 10-4, 10-3), x(0) (0.3, 0.1,0.2),

a "sheet" type dynamics takes place, whereby asmall set of points exist in one dimension (x3) to-wards which trajectories are attracted (stable per-iodic movement), while undergoing nonattractor

bound chaotic behavior along the other two di-mensions (xl, x2).

Thus, one or two stocks exhibit some periodicmotion, referred to as "calm" dynamics, whilethe other (two or one stock respectively) are un-

dergoing isolated or limited turbulent dynamics.These examples demonstrate another dimensionof the universality of the map in depicting a vari-ety of qualitatively different dynamics, a subjectthat could be of interest to the contributors ofthis journal.Another topic, partly related to the cases dis-

cussed above and central to the objective of thisjournal, is the manner in which the generality ofthis map can be employed in the natural sciencesas well. In Fig. 3, an example is given of howthis three-stock two-location statement of thegeneral stock-location interaction map can, un-der chaotic regimes without any recognizable at-tractors in phase space, generate inside the unitcube distributions reminiscent of the manner inwhich certain cosmological models envision visi-ble matter in the universe to be distributed over

FIGURE 3 Chaotic dynamics in the unit cube in absence ofattractors; 25,000 iterations. See text for parameter specifica-tions.

CHALLENGES IN SOCIO-SPATIAL SCIENCE 15

three-dimensional space. Figure 3 is generatedunder the specifications:

[a]-1

0 -2

0 0

0

0

0

[b]0 0 10 -2 0

0 0 -2

A (10-4, l0-4, 10-6), x(0) (0.3, 0.1,0.2).

6 CONCLUSIONS

enforce some social scientists fears that socialtheorizing is (and should be regarded accordinglyas) much more of an individual activity ratherthan a collective (public) good.

Nonlinear interactive dynamics modeling inthe socio-spatial sciences presents an excellentforum for such a resolution to be abjudicated.The task ahead is to show beyond a reasonabledoubt that socio-spatial systems do indeed con-tain chaos rather than X&ocr.

Acknowledgement

The author wishes to thank his research assistantTerry Zhao, who in the Fall of 1992 producedthe computer graphics shown in this paper.

Mathematical modeling in the social sciences hasbeen in place for about a century now. Beyondacting as an intellectual pilot, mathematical mod-eling must move ahead and show its capacity tobecome the heavy duty container of commodities.If this is not achieved in short order, doubtsregarding the sustainability of such an intellectualactivity will increase. These doubts will furthererode the credibility base of social sciences, andcertainly it will impact upon its competitivenessin being supported by the collective resources ofhuman societies. A failure of this magnitude will

References

Chaos, Solitons and Fractals (1994). Special Issue on nonlineardynamics in urban and transportation analysis, 4(4).

J. Cohen and I. Stewart (1994). The Collapse of Chaos, Vik-ing, New York.

D.S. Dendrinos and M. Sonis (1990). Chaos and Socio-SpatialDynamics, Springer-Verlag, New York.

S.H. Kellert (1993). In the Wake of Chaos, University ofChicago Press, Chicago.

L.D. Kiel and E. Elliott (1995). Chaos Theory in the SocialSciences; Foundations and Applications, Michigan Univer-sity Press, Ann Arbor.

M. Sonis and V. Gontar (1992). Entropy principle of extrem-ality, Socio-Spatial Dynamics, 3(2), 61-73.

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