# Chaos Theory and Complexity Theory

## Abstract and Keywords

Chaos theory and complexity theory, collectively known as nonlinear dynamics or dynamical systems theory, provide a mathematical framework for thinking about change over time. Chaos theory seeks an understanding of simple systems that may change in a sudden, unexpected, or irregular way. Complexity theory focuses on complex systems involving numerous interacting parts, which often give rise to unexpected order. The framework that encompasses both theories is one of nonlinear interactions between variables that give rise to outcomes that are not easily predictable. This entry provides a nonmathematical introduction, discussion of current research, and references for further reading.

Keywords: agent-based models, chaos theory, complexity theory, dynamical systems, nonlinear dynamics, system dynamics

Since the late 1960s, chaos theory and complexity theory have grown out of a variety of fields, including physics, biology, and social science, in which researchers grapple with systems in which multiple variables interact, affect each other, and change through time (Bak, 1997; Ball, 2004; Gleick, 1987; Hudson, 2000; Sprott, 2003; Strogatz, 1994; Waldrop, 1992; Warren, Franklin, & Streeter, 1998). General systems theory, which, with its focus on feedback and dynamics, is familiar to social workers, is an important precursor of both (Ashby, 1962; Boulding, 1956; Sterman, 2000; Von Bertalanffy, 1968).

Collectively known as dynamical systems theory or nonlinear dynamics, neither chaos nor complexity theory is a theory in the ordinary sense of the word. Neither attempts to explain a specific phenomenon. Rather, each is a collection of mathematical and computer models and empirical techniques aimed at understanding the way in which systems change through time. Chaos theory addresses simple feedback systems of a small number of variables that nevertheless show complicated and often unpredictable behavior (Briggs & Peat, 1990; Gleick, 1987; Kaplan & Glass, 1995; Peak & Frame, 1994). Complexity theory seeks an understanding of the ways in which large, complex systems emerge from local interactions, as well as the ways in which they change and develop over time (Bak, 1997; Ball, 2004; Barabasi, 2003; Kohler & Gumerman, 2000; Resnick, 1994; Strogatz, 2003). Together, they offer a lens through which social scientists have examined human systems including individuals (Hufford, Witkiewicz, Shields, Kodya, & Caruso, 2003; Warren, 2002), interacting dyads (Gottman, Swanson, & Swanson, 2002; Warren, Newsome, & Roe, 2004), small groups (Pincus & Guastello, 2005), schools (Warren, Craciun, & Anderson-Butcher, 2005), neighborhoods (Fossett, 2006; Schelling, 1971), geographic regions (Parker, Berger, & Manson, 2002), societies (Gunduz, 2000), and even international systems (Saperstein, 1996; Turchin, 2003).

Theoretical Foundations

## Dynamics

Dynamical systems are simply those that move or change over time (Williams, 1997). At least since Heraclitus’s comment that no one can put a foot in the same river twice—the second time it is not the same river, and she or he is not the same person—there has been no doubt that all social systems are dynamic (Haxton, 2001). The systematic study of processes of change is known as dynamics (Sprott, 2003; Strogatz, 1994; Williams).

## Nonlinearity

Anyone who remembers high-school algebra or introductory statistics remembers the form of a linear equation *y* = *mx* + *b*, the equation for a straight line, in which *m* represents the constant slope, *b* is the intercept, and *x* and *y* are variables. On the other hand, as Williams (1997) states, “A nonlinear equation is an equation involving two variables, say x and y, and two coefficients, say b and c, in some form that doesn’t plot as a straight line on ordinary graph paper” (p. 9). Of course, nonlinear equations may involve more than two variables. Typical nonlinear equations involve multiplying two different variables or raising a variable to some power (Sprott, 2003; Strogatz, 1994).

Scientists use mathematical equations to describe the world, and nonlinear equations often provide a better description. For instance, if one student studies for 5 hours per week and another for 10 hours per week, most people would expect the second student to outperform the first, all other things being equal. But if one student studies for 40 hours a week and the other for 45, would the difference be as great? Would there be any improvement at all? Eventually, the student reaches the limits of her capacity—and, of course, she cannot get better than an A+. A linear equation, which implies the same change in grade for any change in study time, cannot capture such ceiling effects. A nonlinear equation, on the other hand, implies that change in the effect will vary over the range of the cause (Williams, 1997). A nonlinear equation therefore implies that the effect is not necessarily proportional to the cause (Pryor & Bright, 2003).

## Nonlinear Dynamics

Nonlinear dynamics involve nonlinear change over time. For instance, population growth often starts out slowly, speeds up, and then levels off as resources are exhausted, a pattern known as logistic growth (Kaplan & Glass, 1995). Logistic growth is quite common, applying to situations ranging from increasing animal populations to construction of subway systems (Modis, 1992). Other processes for which there is some evidence of nonlinearity include human development, aggressive behavior, and substance abuse (Hufford et al., 2003; Thelen & Smith, 1994; Warren et al., 2004).

Nonlinear dynamics give rise to surprising behaviors. Two of these behaviors have been of particular interest to social scientists. Bifurcations involve sudden changes in the state of a system (Strogatz, 1994). For instance, it is possible for systems to show a sudden increase or decrease in values. Such changes are known as “catastrophes” (Strogatz, 1994, p. 69; see also Guastello, 1995). There is evidence that relapse into substance abuse can best be modeled as such a sudden change (Hufford et al., 2003). Bifurcations can also involve change from a stable equilibrium to an oscillation (Strogatz, 1994). Such a bifurcation could explain volatility in human behaviors such as aggression (Warren et al., 2004).

In some cases, a series of bifurcations can lead to deterministic chaos, after which chaos theory is named (Gleick, 1987; Sprott, 2003; Williams, 1997). Deterministic chaos is an oscillation that never repeats itself (Peak & Frame, 1994). Chaotic systems have the interesting property that any change in value, no matter how small, will tend to grow over time (Gleick, 1987; Strogatz, 1994). This is known as sensitive dependence on initial conditions and, more colloquially, as the butterfly effect (Gleick; Sprott, 2003). The point of the latter phrase is that the flap of a butterfly wing in the Amazon rain forest can, in theory, eventually change the path of a tornado in Texas (Lorenz, 1996). This idea has worked its way into popular culture and has formed the basis of several movies, including * Sliding Doors* (1998) and

*(1998). It has profound and disquieting implications for prediction, one of the traditional goals of science (Smith, 2007).*

*Run, Lola, Run*Whether linear or nonlinear, dynamical systems frequently move toward an attractor. This could be a point or a cycle that values tend to approach regardless of where they start out (Strogatz, 1994). Because chaotic systems do not repeat their values, their attractor is neither a point, which would effectively give the same value over and over again, nor a simple cycle, which would give a series of repeated values. Rather, their attractors include an infinite number of possible cycles within a limited range. Such attractors are known as strange attractors (Gleick, 1987; Sprott, 2003; Strogatz). Strange attractors are typically fractals, the term for objects that have a fractional dimension (Gleick; Strogatz).

## Complexity, Connection, and Emergence

Chaos theory enlightened scientists to the surprising, complex behavior that simple systems of a few interacting variables can show (Gleick, 1987). From the 1970s, and intensifying greatly in the 1980s, a concerted effort was underway to apply ideas from nonlinear dynamics to complex systems, which might have dozens, hundreds, or even thousands of variables characterized by nonlinear interactions (Cowan, Pines, & Meltzer, 1994; Lewin, 1993; Schelling, 1978; Sterman, 2000; Waldrop, 1992).

There are several possible ways to do this. One is to use nonlinear equations that are similar to those used in chaos theory, but to use more of them. Not surprisingly, this has become easier with the advent of powerful desktop computers. The use of computers to construct visual models of dynamic equations shown as feedback loops between variables, known as system dynamics, has been particularly influential because of its flexibility and ease of use (Hovmand, 2003; Sterman, 2000). Robards and Gillespie (2000) have suggested that the traditional social-work educational emphasis on systems be revised to include system dynamics modeling.

A second way of grappling with complex systems is to treat them as collections of interacting agents, rather than interacting variables. These agents might represent people, but they might also represent automobiles, primates, insects, or even grains of sand (Bak, 1997; Fossett, 2006; Kohler & Gumerman, 2000; Resnick, 1994). Each agent has a set of rules that govern how it interacts with others. Interactions could occur within a particular geographic space, or they could occur on a social network or through set interactions between the agents (Axelrod, 1997; Kohler & Gumerman). Order in the form of large-scale social structures frequently emerges from the decentralized interactions of the agents, a process known as emergence (Fossett; Holland, 1998; Kohler & Gumerman). The emergent order, in turn, constrains the actions of the agents (Fossett; Resnick).

A classic example of this approach is the model of segregation developed by Nobel Prize–winning economist Thomas Schelling. The agents could represent any two groups, but they are typically demarcated by color, most obviously black and white. Agents prefer that some percentage of their neighbors be of their own color. That is the only rule in the model, which consistently produces two results. The first is that agents end up living with a much larger percentage of neighbors of their own color than the minimum that they would prefer (Resnick, 1994). The second is that segregation locks in because any agent that breaks the pattern will move into a neighborhood with fewer of its own color than it prefers. Thus, segregation arises through interactions between agents, and once it has arisen it constrains the actions of the agents (Chen, Irwin, Jayaprakash, & Warren, 2006; Fossett, 2006; Resnick; Schelling, 1971, 1978).

Because one can treat any set of interactions between individuals as a social network, social network analysis makes an intuitively appealing framework for the statistical analysis of complex systems. For many years social network analysts focused on relatively enduring social network structures (Wasserman & Faust, 1994). However, recent years have seen an explosion of interest in social network dynamics, and social network analysis is becoming an increasingly potent tool for the study of the dynamics of complex systems (Butts, 2009; Kossinets & Watts, 2006; Ripley, Snijders & Preciado, 2012).

Applications in Social Science and Social Work

Application of nonlinear dynamics to social science has grown apace (Vallacher, Read, & Nowak, 2002), and authors have begun to call for the application of nonlinear dynamics to social work (Green & McDermott, 2010; Hudson, 2010). The remainder of this entry will review some applications of potential relevance to social workers.

## Individual Processes

Dynamical systems theory has led to an increased focus on human processes, particularly those that involve fluctuation, irregularity, or sudden change (Vallacher et al., 2002). Simply acknowledging the fluctuations and capacity for self-organization of human systems can be of considerable value. For instance, Pryor and Bright (2003) have developed a “chaos theory of careers,” which “seeks to understand individuals as complex dynamical, nonlinear, unique, emergent, purposeful open systems existing and interacting with an environment comprising systems with similar characteristics” (p. 123). At least one randomized experiment supports the effectiveness of their intervention (McKay, Bright, & Pryor, 2005). Witte, Fitzpatrick, Warren, Schatschneider, and Schmidt (2006) quantified the variability of suicidality in a sample of undergraduates and found evidence that those with previous suicide attempts show increased variability. Warren and Knox (2000) and Warren (2002) applied a piecewise linear model, in which two straight lines are separated by a threshold (Tong, 1993), to time series of problem behaviors of sex offenders. Both studies found evidence of nonlinearity that could lead to extreme variability and cycling in these behaviors.

Regular fluctuations, known as oscillations, have been the subject of several studies. Bisconti, Bergeman, and Boker (2004) applied a differential equation, which predicts the rate of change over time, to the emotional well-being of recently bereaved widows. The equation predicted that emotional well-being would oscillate. Their statistical analysis found a cycle lasting about 47 days on average, but also found that it flattened after about 98 days. One study has modeled relapse in substance abuse as a bifurcation in which current consumption is a nonlinear function of past consumption; beyond a certain level of past drinking, current drinking gets suddenly worse, a situation known as a cusp catastrophe (Hufford et al., 2003).

## Dyadic Processes

Interactions between two people are obviously important in marriage, and in recent years researchers have developed several approaches to studying the dynamics of such interactions. A team of psychologists and mathematicians led by John Gottman et al. (2002) has developed a nonlinear model of the influence that spouses have on each other. Influence can be either positive or negative, with negative influence happening more quickly and easily than positive influence. Each spouse can influence the other, and this pattern can produce attractors in which each influences the other, either positively or negatively. These patterns are evaluated by observing a 15-minute conversation between the couple. They have proven to have predictive power and Gottman’s group has begun designing interventions based on them, in some cases with success (Gottman et al.).

## Therapeutic Groups and Milieus

Authors have long noted that the life cycle of therapy groups shows periods of rapid organization, and have even proposed the possibility of mathematical models of group interactions (Lewin, 1951; Yalom, 1985). Dynamical systems theory, with its focus on rapid change and emergence of order through decentralized interactions, appears to be a promising framework for the study of groups, and a number of authors have called for the use of dynamical systems theory in understanding group interventions (Fuhriman & Burlingame, 1994; Rubenfeld, 2001). Empirical work has begun to yield a dynamical understanding of group processes. For instance, Pincus and Guastello (2005) found complex but coherent patterns of turn taking in a six-member therapy group for adolescent sex offenders.

Since the 1980s there has been the development of an immense literature on the ways in which interactions lead to cooperation in groups. This literature began with Axelrod and Hamilton’s (1981) seminal work on the maintenance of cooperation through reciprocity—you scratch my back, I’ll scratch yours—and includes work on the maintenance of cooperation through reputation (Nowak & Sigmund, 2005)—you scratch someone else’s back, I’ll scratch yours. This literature offers the possibility of a fruitful theory for the maintenance of cooperation in both small groups and treatment milieus such as therapeutic communities for substance abuse. Doogan, Warren, Hiance, and Linley (2010) modeled the interactions of therapeutic community residents and then compared the model outcomes with clinical data from a 90-bed therapeutic community. The results of the comparison suggested that direct reciprocity was unable to explain cooperation between program residents and that reputation effects might offer a better explanation.

## Organizations

The study of organizations is another area in which researchers apply dynamical systems theory (Guastello, 1995). The system dynamics approach to the study of complex systems has largely been developed for the modeling of organizations (Sterman, 2000).

System dynamics–based studies of organizations are appearing in the social-work literature. For example, Cho and Gillespie (2006) have analyzed the feedback relationship between government and social-service agencies, concluding that the feedback mechanisms are sufficiently complex so that government regulations can inadvertently lower the quality of services that they are meant to improve. Further, the time lag between government intervention and results in the community is sufficiently long that regulators may be tempted to intervene too frequently.

Organizations not only interact with funders; they also interact with each other, sharing such resources as staff or space although they may compete in other ways. Using social network analysis, Bunger, (2012) was able to demonstrate that the perceived trustworthiness of organizational partners moderates the relationship of competition and willingness to share resources. Reputation therefore appears to explain cooperation among organizations as well as individuals. Wulczyn et al. (2010) have proposed that child protection services should be reconceptualized in terms of complex systems.

Conclusion

Chaos and complexity theory address important aspects of human experience, particularly sudden change, fluctuations, and the ability of groups and organizations to self-organize. Applied social science based on dynamical systems theory is no longer a rarity. However, those social-work researchers and social workers who wish to apply dynamical systems theory face at least three challenges. The first is theoretical. How do we connect social-work theory with the mathematical models that underlie chaos and complexity theory? Current practice ranges from the frankly metaphorical to the development of complex nonlinear differential equations. The second is empirical. How do we find the data with which to test our models? It seems clear that improved observational techniques are leading to progress in this area (Gottman et al., 2002; Joy-Bryant, 1992; Pincus & Guastello, 2005). The third is pragmatic. How do we integrate dynamical systems theory into social-work practice? Much of this integration may simply involve a greater appreciation for the role of sudden change, fluctuations, and self-organization. But clinicians are beginning to develop interventions based on dynamical systems theory that could have an enormous influence on social work and other helping professions in the future (Gottman et al.; Pryor & Bright, 2003).

Further Research Literature

Smith (2007) offers an excellent introduction to the basics of chaos theory for individuals with no mathematical background beyond high-school algebra. Peak and Frame (1994) also offer an excellent nonmathematical introduction to chaos theory and touch on such complexity theory concerns as emergence. For those who have at some point studied introductory calculus, Williams (1997) has written an excellent introduction with extensive discussion of the basics of empirical work. Kaplan and Glass (1995) also provide an excellent introduction that includes a discussion of emergence. The first half can be read with only a background in algebra, whereas the second half requires some calculus. Strogatz (1994), author of the standard introductory undergraduate text, does assume some background in differential equations. Sprott (2003) gives an encyclopedic treatment of chaos theory, with a well-written text that makes the highly mathematical treatment easier to follow. Resnick (1994) gives a thought-provoking introduction to agent-based models and Kohler and Gumerman (2000) offer numerous applications to human systems. Vallacher et al. (2002) provide a useful discussion of the place of dynamical systems theory in psychology, whereas Gottman, Murray, Swanson, Tyson, and Swanson (2005) offer a model of marriage and marital therapy based on nonlinear differential equations.

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## Further Reading

Center for Connected Learning and Computer-Based Modeling: http://ccl.northwestern.edu

New England Complex Systems Institute: http://necsi.edu/

Repast Suite: http://repast.sourceforge.net/

Santa Fe Institute: http://www.santafe.edu/

Society for Chaos Theory in Psychology & Life Sciences: http://www.societyforchaostheory.org

Sprott’s Gateway: http://sprott.physics.wisc.edu

System Dynamics Society: http://www.systemdynamics.org/