More Resources for
"An Introduction to Graph
Theory and Complex
Networks" by Maarten van
Steen (2010; 287p., 6MB). http://pages.di.unipi.it/ricci/book-watermarked.pdf
"Complex Network Theory: An
Introductory Tutorial" by Ashish
Anand (2013, 120 slides, 6.8MB).
More about network structures
and network analysis, by Matthew
Denny, Institute for Social
Science Research, University of
Massachusetts, Amherst. [PDF]
More publications, people,
More resources for Applications -- Paradigm & Book Lists
Nonlinear Dynamics, Psychology, and Life Sciences
is the refereed research journal of the Society for Chaos Theory in Psychology & Life Sciences. Since its inception in 1997, NDPLS is the only refereed research journal that is uniquely devoted to this range of nonlinear applications and related methodologies. See the journal's home page
for contents, data base indexing, citation, editorial, and related information.
This special issue of Nonlinear Dynamics, Psychology, and Life Sciences (January, 2007) was devoted to the paradigm question as it was manifest in a variety of disciplinary areas. See contents.
The Impact of Edward Lorenz. Special issue of NDPLS (July, 2009) pays a historical tribute to Lorenz discovery of the butterfly effect, its mathematical history and later developments, and its applications in economics, psychology, ecology, and elsewhere. See contents. Special order this issue.
The Nonlinear Dynamical Bookshelf is a regular feature of the SCTPLS Newsletter (send to active members) that presents a nnouncements and brief summaries of new books on topics related to nonlinear dynamics. Contents are limited to information we can collect from book publishers or that crawl into our hands by any other means.
Open access book reviews: In an effort to help the world get caught up on its reading, NDPLS has made its book reviews published since 2004 free access on its web site. Browse the journal's contents to see the possibilities.
Books written by members of the Society for Chaos Theory in Psychology & Life Sciences. This list is as complete as we can get it for now, and it is updated regularly. Most are technical in nature. Some of these works go beyond the scope of nonlinear science. Some are whimsical. All are recommended reads.
Recent books by members is a sub-list of the above that starts in 2009.
More resources for
Chaos and Complexity in Psychology: The Theory of Nonlinear Dynamical Systems edited by Guastello, Koopmans, & Pincus (2009). This collection captures the state of the science of nonlinear psychology in application areas ranging from neuroscience to organizational behavior. For further description and ordering information see Cambridge University Press.
Article: "Chaos as a Psychological Construct: Historical Roots, Principal Findings, and Current Growth Directions" by S. J. Guastello
Chaos, Complexity, and Creative Behavior. Special issue of NDPLS (April, 2011) explores nonlinear dynamics of the cognitive, process, product, and diffusion aspects of creative behavior. See contents. Inquire about availability.
Chaos, Complexity, and Creative Behavior. Special issue of NDPLS (April, 2011) explores nonlinear dynamics of the cognitive, process, product, and diffusion aspects of creative behavior.
Developmental Psychopathology. Special issue of NDPLS (July, 2012) examines parent-child interactions from a dynamical point of view. See Contents. Inquire about availability.
Nonlinear Dynamics in Education. Special issue of NDPLS (January, 2014) examines parent-child interactions from a dynamical point of view. See Contents. Special order this issue.
Interpersonal Synchronization. Special issue of NDPLS (April, 2016) examines a fast-moving research area about how body movements, autonomic arousal, and EEGs synchronize between dyads, such as therapy-client dyads, and larger work teams, and the effect synchronization has on various outcomes. See contents. Special order this issue.
Article: "Nonlinear Dynamics in Psychology" by S. J. Guastello. This open access article from Discrete Dynamics in Nature and Society, vol. 6, pp. 11-29, 2001 gives an overview of applications in psychology, except neuroscience, as they existed through early 2000.
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Optimum Variability. Special issue of NDPLS (October, 2015) examines the relationship system complexity and the health, functionality, and adaptability of biomedical systems, individual well-being, and work group dynamics. See contents
. Special order this issue.
Medical Practice. Special issue of NDPLS (October, 2010) offers theoretical and empirical studies that indicate that a paradigm shift in neurology, cardiology, rehabilitation, and other areas of medical practice is very necessary. See contents. Special order this issue.
Brain Dynamics. Special issue of NDPLS (January, 2012) explores developments in brain connectivity and networks as seen through temporal dynamics. See contents. Special order this issue.
Psychomotor Coordination and Control. Special issue of NDPLS (Jan. 2009) explores developments in psychomotor learning and skill acquisition and applications to rehabilitation. See contents. Special order this issue.
Handbook on Complexity in Health, edited by J. P. Sturmberg & C. M. Martin (2013) offers over 1000 pages of viewpoints and research on medical thinking and practice, behavioral medicine and psychiatry from the perspective of nonlinear dynamics and complex systems. See table of contents.
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Nonlinear Organizational Dynamics. Special issue of NDPLS (January, 2013) presents empirical studies on individual work performance, group dynamics, organizational behavior. See Contents
. Special order this issue.
Article: "A Complex Adaptive Systems Model of Organizational Change," by K. J. Dooley.
Article: "Nonlinear dynamical systems for theory and research in ergonomics," by S. J. Guastello.
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Economics & Policy
Complexity in economics edited by J. B. Rosser, Jr. The international library of critical writings in economics 174. UK: Edward Elgar. This 3-volume set (2004) compiles a wide range of important and fundamental works on nonlinear dynamics in all the areas of economics. Further description and ordering information.
Article: "Complexity and Behavioral Economics" by J. B. Rosser, Jr. & M. V. Rosser (2015). The works of H. Simon figure prominently in this survey-review article.
Econophysics Research by V. Yakovenko
Attractors are the elements of nonlinear dynamics. An attractor is a piece of space. When an object enters, it does not exist unless a substantial force is applied to it. The simplest attractor is the fixed point. Some fixed points have spiral paths and some are more direct. Limit cycles and chaotic attractors are more complex in their movements over time, but they have the same level of structural stability. Structural stability means that all objects in the space are moving around according to the same rules.
Oscillators, also known as limit cycles, are another type of attractor. Like the moon revolving around the early, once an object gets too close to the limit cycle it continues to orbit indefinitely or until a substantial force is applied. Oscillators can be pure and simple, or dampened to a fixed point by means of a control parameter. They can also be perturbed in the opposite direction to become aperiodic oscillators. There is a gradual transition from aperiodic produce a chaotic time series.
A control parameter is similar to an independent variable in conventional research. Here it has the effect of altering the dynamics of the order parameter, which is similar in meaning to a dependent variable, except that it is not necessarily dependent -- it operates on its own intrinsic dynamics.
Repellors are like attractors, but they work backwards. Objects that veer too close to them are pushed outward and can go anywhere, so long as they go away. This property of an indeterminable final outcome is what makes repellors unstable. Fixed points and oscillators, in contrast, are stable. Chaotic attractors (described below) are also stable in spite of their popular association with unpredictability.
A saddle has mixed properties of an attractor and a repellor. Objects are drawn to it, but are pushed away once they arrive. An example is the perturbed pendulum shown at the right. A saddle is also unstable.
Bifurcations are splits in a dynamic field that can occur when an
attractor changes from one type to another, or where different
dynamics are occurring in juxtaposing pieces of space. They
can even produce the appearance and disappearance of new
attractors. Bifurcations are patterns of instability that can be as
simple as a critical point, curved trajectories, or more complex in
structure. One or more control parameters is often involved to
change the system from one regime to the next.
The logistic map is one of the classic bifurcations. It was first
introduced to solve a problem in population dynamics and has
seen a variety of applications since then. Start with the function
= CX1 (1 - X1 )
, with X1
in the range between 0 and 1.
and run it through the equation again to produce X3
repeat a few more times. When C
small, the results stay within a steady
state. When C
becomes somewhat larger
(let it become gradually larger
than 1.0), X
goes into oscillations.
becomes larger still, the
become more complex (period doubling). When C4
, the output is
The chaotic regime is rendered as a jumble of interlaced trajectories. The
vertical striations are intentional. There are brief episodes of calm within
Bifurcations are often experienced as critical points or tipping points.
are a critical feature of catastrophe models.
Chaos is a particular nonlinear dynamic wherein seemingly random events
are actually predictable from simple deterministic equations. Thus, a
phenomenon that appears unpredictable in the short term may indeed be
globally stable in the long term. It will exhibit clear boundaries and display
sensitivity to initial conditions. Small differences in initial states
compound to produce markedly different end states later on in time. The
latter property is also known as The Butterfly Effect and was first discovered
by Edward Lorenz during an investigation of weather patterns.
Chaos has some important connections and relationships to other dynamics,
however, such as attractors, bifurcations, fractals, and self-organization.
all examples of chaos are chaotic attractors, but several dozen structures for
chaotic attractors are known, however. A chaotic attractor has the stability
characteristic of the simpler attractors that were already described because
all the points within the attractor and moving according to the same rule.
The movements within the attractor basin are chaotic, and they contain
both expanding and contracting movements. The paths of motion expend
to the outer rim, then back again toward the center -- a pattern that can be
observed in schools of fish or flocks of birds.
The famous Lorenz attractor is shown at the right. A point moves along any
of the trajectories on one lobe, but suddenly switches to the other
lobe. The transition from one lobe to the other is as random as
flipping a coin -- with the exception that it is a fully deterministic
process instead. The Lorenz attractor is a system of three order
paramters (movements along X, Y, and Z cartesean coordinates)
and three control paramters. The control parameters goven the
amount of spread between the lobes and their orientation along
the cartesean axes.
There are a few pathways by which a non-chaotic system can
become chaotic. One is to induce bifurcations, which we have
already described with the logistic map.
Another pathway is to couple oscillators together. One oscillator
acts as a control parameter that drives another oscillator.
A third option is to create a field with multiple fixed point
attractors close together and send an object into the field. The
latter example actually captures the three body problem that
Henri Poincare was studying in the 1890s when he first discovered
what we call chaos today. The word chaos did not appear in the
system science vocabulary until the 1970s, however.
A fractal is a geometric form in a non-integer number of dimensions, meaning that they do not fill up a whole 2-D or 3-D space. Fractals also have self-repeating structures. The same overall pattern is visible if we zoom in or out to different levels of scale. Their essential structures can be found in many examples in nature - the shapes of snowflakes, vegetable, lightning, neural structures. Why are they so visually engaging?
J. C. Sprott's Fractal of the Day website, shown at the right, changes every day at midnight, US Central Time. It deploys two basic algorithms. One selects a fractal structure, and the second evaluates the design for its level of complexity and other aesthetic properties. Please visit the archives, which can be reached by tracing the link. Fractal analysis can also be used to assess and compare the complexity of visual images such as abstract art works.
One of the more practical fractal functions is diffusion limited aggre-gation. This concept has been used to map the flow of contagious diseases across physical space. The pathogen spreads quickly down a central pathway, then fans out in multiple directions, then dissipates. It is not too easy to apply this principle, however, when airplanes are involved as carriers (which defies the implicit principle of physical proximity).
There is an important connection between fractal structure and chaos: The basin, or outer boundary of a chaotic attractor is a fractal. This discovery quickly led to the calculation of a fractal dimensions in time series data, which were in turn used to characterize the complexity of a time series of biometric or psychological data. In principle, it should be possible to find a fractal structure at one level of scale that repeats at other (finer, broader) levels of scale.
At one point in history, it was thought that the presence of a fractal structure in a time series was a clear indication that the time series was chaotic. This assumption turned out to be an oversimplification, however. A chaotic time series is composed of expanding and contracting segments. A much better "test for chaos" is the Lyapunov exponent associated with the time series.
A Lyapunov exponent is actually a spectrum of values that is computed from the sequential differences numbers in a time series. A positive exponent indicates expansion, and a negative value indicates contraction toward a fixed point. A perfect 0.0 indicates a pure oscillator, which, in practice could be a little bit purturbed in the chaotic direction or dampened in the direction of a fixed point. The decision about the character of a time series is drawn from the largest Lyapunov exponent, which should be positive, while the sum of the other values should be negative.
Conveniently, the largest Lyapunov exponent can be converted to a fractal dimension. Fractal dimensions between 0 and 1.0 indicate gravitation toward a fixed point. A value of 1.0 could indicate either a line or a perfect oscillator. Values between 1.0 and 2.0 are usually interpreted as the range of self-organized criticality, which reflects a balance between order and chaos. The connection between fractal structures, self-organization, and emergent events, which is developed later on in conjunction with self-organization.
Fractal dimensions between 2.0 and 3.0 are chaotic, but with a bias toward relatively small fluctuations over time that are perforated by a few large ones. An example would be the flight path of a bird of prey that is checking out its terrain suddenly swoops down to the group to check out something delicious. Humans adopt a similar pattern in a grocery store, probably without thinking about it. The caveat here, however, is that grocery stores are much more organized than the critters running through the woods. The grocery store wants us to find what we are looking for; the critters do not want to be found by the hawk.
Fractal dimensions greater than 3.0 are chaotic. One more caveat, however, is that there are many well-known chaotic attractors with fractal dimenions closer to 2.0. Not all examples of chaos are chaotic attractors.
FRACTAL OF DAY LIVE FEED HERE
Fractal of the Day
A new fractal appears every night at midnight CST
Catastrophes are sudden changes in events; they are not necessarily bad or unwanted events as the word "catastrophe" (in English) might suggest. Catastrophe models contain combinations of attractors, repellors, saddles, and bifurcations. According to the classification theorem developed by Rene Thom, all discontinuous changes of events can be described by one of seven elementary topological models. The models are hierarchical such that the simpler ones are embedded in the larger ones.
The simplest model is the fold catastrophe. It describe transitions
between a stable state (attractor) and an unstable state. The shift
between the two modalities is governed by one control parameter
(aka independent variable). When the value of that parameter reaches
a critical point, the system moves into the attractor state, or out of
it. Each catastrophe model contains a bifurcation set. In fold model,
the bifurcation set consists of a single critical point.
The catastrophe models are polynomial structures. The leading
polynomial for the fold response surface is a quadratic term, such
that f(y)/dy = y2 - a, in which a is the control parameter and y is the
observed behavior variable (aka, the dependent measure).
The catastrophe models also have a potential function, which char-
acterizes the behavior of agents that are acting within the model as
positions rather than velocities. In other words, to represent a velocity
we would say f(y)/dy or dy/dt (t = time). To represent the potential
function we would integrate the response surface function; thus for a
fold, f(y)= y3 - ay.
The cusp model is the second-simplest in the series - just complex
enough to be very interesting and uniquely useful. In fact it is over-
whelmingly the most popular catastrophe model in the behavioral
sciences. The cusp requires two control parameters, asymmetry and
bifurcation. To visualize the dynamics, start at the stable state on the
left and follow the outer rim of the surface where bifurcation is high.
If we change the value of the asymmetry parameter, nothing happens
until it reaches a critical point, at which we have a sudden change in
behavior: the control point that indicates what behavior is operating
flips to the upper sheet of the surface. A similar reverse process
occurs when shifting from the upper to the lower stable state.
When bifurcation is low, change is relatively smooth. The cusp point
is a saddle, and is the most instable location on the surface. You
only need to breathe on the object and it moves toward one of the
stable attractor states. The paths drawn in light blue are gradients that
are created by the two control variables. The red spot indicates the
presence of a repellor; comparatively few points land there.
The cusp is often drawn with its bifurcation set, which is essentially a
2-dimensional shadow of the response surface. Therein you can see
the two gradients that are joined at a cusp point. In the application
to occupational accidents shown in the diagram, there were several variables that contributed to the bifurcation parameter. Some had a
negative "swing" to them, and others had a positive "swing", capturing
the contribution of the gradients that are part of the bifurcation
The cusp point is the most unstable point on the response surface. If an
object (agent) is sitting on it, one only needs to breathe on the object to
move it toward one or the other attractor state. The leading polynomial
for the cusp response surface is a cubic term, such that f(y)/dy = y3
– by - a, in which a is the asymmetry parameter and b is the bifurcation
parameter. Its potential function is f(y) = y4 - by2 – ay.
The swallowtail catastrophe model shows movement along a 4-
dimensional response surface that must be shown in two 3-D sections.
The leading polynomial for its response surface is a quartic polynomial:
f(y)/dy = y4 - cy2 - by - a. When the asymmetry parameter, a, is low
in value, objects on the surface can move from an unstable state to a
more interesting part of the surface (shown in the upper portion of the
figure to the right) where the stable states are located. The bifurcation
parameter, b, determines whether points will move from the back of
the surface to the front regions where the stable states are located.
Points can jump between the two stable states, or they can fall through
a cleavage in the surface back to the unstable state (low a). The bias
parameter, c, determines whether a point reaches one or the other stable
To learn more about the cusp and other catastrophes, please visit The
Catasteophe Teacher by Lucien Dujardin. Briefly, however, the butterfly
catastrophe describes movement along a 5-dimensional response
surface. It contains three stable states with repellors in between. Points
of objects can move between adjacent states in cusp-like fashion, or
between disparate states in a more complex fashion.
The last three catastrophes belong to the umbilic catastrophe group.
They are disguished by having 2 dependent measrues (or order
parameters). The wave crest (or hyperbolic umbilic) model consists of
two fold-like variables that are controlled in part by the same bifurcation
parameter. Each behavior has its own asymmetry parameter.
The hair (or eliptic umbilic) model has similar properties as the wave
crest, with the important addition that there is an interaction between
the two dependent variables. It gets its name from its bifurcation
set, which depicts three trajectories coming together at a hair-thin
intersection then fanning out again.The mushtoom (or parabolic umbilic) model has one dependent measure
that follows cusp-like dynamics between two stable states and one
dependent measure that follows fold-like dynamics. The model contains
four control parameters, and there is an interaction between the two
A system that is in a state of chaos, high entropy, or far-from-equilibrium
conditions would exhibit high-dimensional changes in behavior patterns
over time, but not indefinitely so. Systems in that state tend to adopt new
structures that produce Self-organization is sometimes known as "order for
free" because systems acquire their patterns of behavior without any input
from outside sources.
There are four commonly acknowledged models of self-organization:
synergetics, introduced by Herman Haken; the rugged landscape, which
was introduced by Stuart Kauffman; the sandpile, introduced by Per Bak;
and multiple basin dynamics, introduced by James Crutchfield. What they
all have in common is that the system self-organizes in response to the
flow of information from one subsystem to another. In this regard the
principles build on the concepts of cybernetics that were introduced in the
early 1960s, and John von Neumann’s principle of artificial life: all life can
be expressed as the flow of information.
The basic synergetic building block is the driver-slave relationship, which
can be portrayed with simple circles and arrows. The driver behaves
over time (produces output or information) according to some temporal
dynamic such as an oscillation or chaos. The driver's output acts a control
parameter for to an adjacent subsystem, which one the one hand responds
to the temporal dynamics from the driver and produces its own temporal
output. In the simple case, the driver-slave relationship is unidirectional. In
other cases, such as when effective communication and coordination occur
between two people, the relationships are bidirectional.
A larger system would contain more circles and arrows. What we want to
know, however, is what do the arrows mean? This is where the dynamics
are of paramount importance.
Once patterns form and reduce internal entropy, the structures maintain
for a while until a perturbation of sufficient strength occurs that disrupts
the flow. The system adapts again to accommodate the nuances in some
fashion, either through small-scale and gradual change or a marked
reorganization. The latter is a phase shift. For instance, a person might be
experiencing a medical or psychological pathology that is unfortunately
stable and prone to continue until there is an intervention. The interven-
tion takes some time and effort but the system eventually breaks up
its old form of organization and adopts a new one. The change in the
system is akin to water turning to ice or to vapor, or vice versa. The
challenge is to predict when the change will occur. There is a sudden
burst of entropy in the system just before the change takes place, which
the researcher (therapist, manager) would want to measure and monitor.
An important connection here is that the phase shift that occurs in
self-organizing phenomena is a cusp catastrophe function. Researchers
do not always describe it as such, but the equation they generally use
to depict the process is the potential function for the cusp; the only
difference is that sometimes the researchers hold the bifurcation variable
constant rather than a variable that is manipulated or measured.
The red ball in the phase shift diagram indicates the state the system is
in. In the top portion of the diagram it is stuck in a well that represents
an attractor. When sufficient energy or force is applied, the ball comes
out of the well and with just enough of a push moved into the second
well. In some situations we know what well we're stuck in, but not
necessarily what well we want to visit next. The question of how to form
a new attractor state is a challenge in its own right.
For the rugged landscape scenario, imagine that a species of organism
is located on the top of a mountain in a comfortable ecological niche.
The organisms have numerous individual differences in traits that are
not relevant to survival. Then one day something happens and the
organisms need to leave their old niche and find new ones on the
rugged landscape, so they do. In some niches, they only need one or
two traits to function effectively. For other possible niches, they need
several traits. As one might guess, there will be more organism living in a new 1-trait environment, not as many in a 2-trait environment, and so on. Figure 9 is a distribution of K, the number of traits required, and N the number of organisms exhibiting that many traits in the new environment.
The niches in the landscape are depicted at higher and lower elevation levels, where the highest elevation reflects high fitness for the inhabiting organism, and lower elevations for less fit locations. Organisms thus engage in some exploration strategies to search out better niches. Niches have higher elevations to the extent that there are many forms of interaction taking place among the organisms in the niche. The rugged landscape idea became a popular metaphor for business strategies in the 1990s by a number of authors from the business community. For further elaboration, see Kevin Dooley's linked contribution on rugged landscapes.
For the avalanche model, imagine that you have a pile of sand, and new sand is slowly drizzled on top the pile. At first nothing appears to be happening, but each grain of sand is interacting with adjacent grains of sand as new sands falls. There is a critical point at which the pile avalanches into a distribution large and small piles. The frequency distribution of large and small piles follow a power law distribution.
A power law distribution is defined as FREQ[X] = aXb, where X is the variable of interest (pile size), a is a scaling parameter, and b is a shape parameter. Two examples of power law distributions are shown in the diagram. Note the different shapes that are produced when b is negative compared to when b is positive. When b becomes more severely negative, the long tail of the distribution drops more sharply to the X axis. All the self-organizing phenomena of interest contain negative values of b. The |b| is the fractal dimension for the process that presumably produced them. The widespread nature of the 1/fb relationships led to the interpretation of fractal dimensions between 1.0 and 2.0 as being the range of self-organized criticality.
An easy way to determine the fractal structure of a self-organized process is to take the log of the frequency and plot it against the log of the object size. Then calculate a correlation between the two logs. The regression coefficient is the slope of the line, which is negative. The absolute value of the slope is the fractal dimension.
The multiple basin concept of self-organization also builds
on a biological niche metaphor and attempts to explain how
biological species could cross a species barrier. Imagine there
are several basins, each containing a population of some
sort. The populations stay in their niches while they interact,
change, and do whatever else they do. But the niches are
connected, so that once enough entropy builds up within a
basin, a few of the members bounce out into the adjacent
Multiple basin dynamics can also be found in economics
where, for instance, product designs and product prices
combine to meet distinct market needs. Sometimes, however,
a product producer will jump into another basin. It is an open
question as to how similar the process of jumping basins is to
jumping fitness peaks in the N|K model. Arguably, the multiple
basin scenario is a continuation of the N|K story.
Entropy has been mentioned in conjunction with self-
organizing processes, but without definition until now. The
construct has undergone some important developments
since it was introduced in the late 19th century. In its first
incarnation it meant heat loss. This definition gave us the
principle that systems will eventually dissipate heat and
expire from "heat death." This generalization turned out to be
incorrect for a century later. When statistical physics gelled
in the early 20th century, entropy concerned the prediction
of the location of molecules in motion under conditions of
heat and pressure. It was not possible to target individual
molecules, but it was possible to create metrics of the
average motion of the molecules. This perspective produced
calculations that eventually led to third Shannon entropy and
information functions in the late 1940s.
The Shannon metrics are probably the most widely used versions of entropy today, either directly or as a basis of the derivation of further entropy metrics. Imagine that a system can take on any of a number of discrete states over time. It takes information to predict those states, and any variability for which information is not available to predict is considered entropy. Entropy and information add up to HMAX, maximum information, which occurs when the states of a system all have equal probabilities of occurrence.
The nonlinear dynamical systems perspective on entropy, which is credited to Ilya Prigogine, however, is that entropy is generated by a system as it changes behavior over time, and thus it has become commonplace to treat information and entropy as the same thing and designate them with the same formula: Hs = Σ [p log2(1/p)], where p is the probability associated with an observation belonging to one category in a set of categories; the summation is over the set of categories. Some authors, however, continue to distinguish the constructs of information and entropy as they were originally intended.
Other measures of entropy have been developed for different types of NDS problems, however. A short list includes topological entropy, Kolmogorov-Sinai entropy, mutual entropy, approximate entropy, and the Kullback-Leibler statistic for the correspondence between a model and the data.
To return to self-organizing phenomena, self-organization occurs when the new structure provides a reduction in entropy associated with the possible alternative states of the system. In other words the system picks a state that it likes best, so to speak. The construct of minimum entropy, introduced by S. Lee Hong, or free energy, introduced by Karl Friston, reflect a system's proclivity to adopt a neurological, cognitive, or behavioral strategy that minimizes the number of degrees of freedom required to make a maximally adaptive response.
A related principle is the performance-variability paradox. There is a tendency to think of skilled performance (sports, music, carpentry) as actions produced exactly the same way each time they are produced. There are small amounts of variability, nonetheless, that arise from the numerous neurological and cognitive degrees of freedom that go into producing the action. You can prove the point to yourself by
signing your name six times on a piece of paper. Is each signature
exactly like the others? It is these degrees of freedom that make
an adaptive response and new levels of performance possible.
The complexity range of self-organized criticality reflects a
(living) system's balance between being complex enough to
adapt effectively and minimizing the number of free movements
necessary to do so. Unhealthy systems tend to be overly rigid.
Overly complex systems and behavioral repertoires tend to waste
a lot of energy, which could be detrimental in other ways.
One of the problems that made the idea of complexity famous was that if many agents within a system are interacting simul- taneously, it is impossible to calculate the outcomes of each of them individually and predict further outcomes for other agents with which they interact. Calculating the possible orders in which they could possibly interact would be a daunting task by itself. What is possible, however, is to put the agents into a system and allow them to interact according to specific rules. We can also specify different rules for different agents, in which case we have heterogeneous agents. After the simulation has run long enough, the patterns of interaction stabilize into a self-organized system. The figure from Bankes and Lempert (2004) shows distribution of four types of entities that emerged after a period of time in which their agents interacted.
Agent-based models are part of a family of computational systems that illustrate self-organization dynamics such as cellular automata, genetic algorithms, and spin-glass models. Briefly, cellular automata are agent-based models that are organized on a grid. One cell affects the action of adjacent cells according to some specified rule. The example shown here is very elementary, but it conveys the core idea. The rule structures are chosen by the researchers within the context of a particular problem. The most extensive work in this area is credited to Stephen Wolfram and his New Kind of Science.
Genetic algorithms were first developed to model real genetic and evolutionary processes without having to wait thousands of years to see results. An organism is defined as a string of numbers that represent its genetic code. Organisms then interact in a completely random fashion (or according to other rules specified by the researcher) and "breed." Mutation rules can also be coded into the system. New organisms are then tagged with a fitness index that defines their viability for survival. Ultimately we can see what happens to the computational species relatively soon.
Genetic algorithms have also found a home in industrial design. For instance one can take two or more versions of an object, e.g. an automotive design, characterize them as a string of numbers, and let them breed. The results can be filtered for functionality and usability, and aesthetic properties. The winning possibilities might find their way into real-world production.
Spin-glass models formed the basis of NK or Rugged Landscape models of self-organizing behavior. In principle some of the agents have common properties, and other agents have different common properties. The properties can be complex and mixed up in any theoretical way. After a certain amount of "spinning" together in a closed system, they aggregate into relatively homogenous subgroups.
To learn more about agent-based modeling and to see some examples in action, please visit some of the links included here. The Sugarscape model for artificial societies that was developed by economists at the Brookings Institute is particularly comprehensive for its logical development that closely parallels a real-world economy as rules of interaction are sequentially introduced.
The common use of the word has proliferated in recent years,
but it has a specific, technical origin. Psychologists remember
the maxim from Gestalt psychology, "The whole is greater than
the sum of its parts." The idea originated in scientific venues a
decade earlier, however, with the sociologist Durkheim, who was
trying to conceptualize the appropriate topics for a scientific study
of sociology. The central concern was that sociology needed to
study phenomena that could not be reduced to the psychology of
individuals. The essential solution went as follows: The process starts
with individuals who interact, do business, and so on. After enough interactions, patterns take hold that become institutionalized or
become institutions as we regularly think about them. When the
institution forms, it has a top-down effect on the individuals such
that any new individuals entering the system need to conform
to the demands, which are hopefully rational, of the overarching
system. Emergence comes in two forms, light and strong. In the
light version, the overarching structure forms but does not have a
visible top-down effect. In the strong situations, there is a visible
top-down effect. The dynamics of emergence were captured in
some laboratory experiments by Karl Weick in the early 1970s
on the topic of "experimental cultures." Groups of 3 human
subjects were recruited for a group task. They worked together
until they mastered their routine. Then, one by one, the members
of the groups were replaced by a new person. The replacement
continued until all personnel were changed. At the end of 11
generations, the newest groups followed the same work patterns
as the original group, even though the originators were no longer
part of the system.
Two types of emergence are often observed in live social systems.
One is the phase shift dynamic. The second is the avalanche
dynamic that produces 1/fb relationships. Physical boundaries
have an impact on the emergence of phenomena as well.
Neuroscientists are also investigating the extent to which bottom-
up and top-down dynamics from brain circuits and localization
areas are combining to produce what is commonly interpreted as
The first example of synchronization in mechanical systems was
reported in 1665 by Christiaan Huygens, who noticed that two
clocks that were ticking on their own cycles eventually ticked in
unison. The communication between clocks occurred because
vibrations were transferred between them through a wooden
shelf. Another prototype illustration is synchronization of a
particular species of fireflies, as told by Steven Strogatz: In the
early part of the evening the flies flash on and off, which is their
means of communicating with each other, which they do at their
own rates. As they start to interact, they pulse on and off in
synchrony so that the whole forest lights up and turns off as if one were flipping a light switch. William Strutt opened investigations into the structure of sound waves in 1879, including those that appear synchronized. He observed that two organ pipes generating the same pitch and timbre would negate each other's sound if they were placed too close together. Thus two oscillators could exhibit an inverse synchronization relationship that he called oscillation quelching.
Based on the following century of advancements in the study of oscillating phenomena, Pikovsky et al. defined synchronization as "an adjustment of the rhythms of oscillating objects due to their weak interaction" (Synchronization: A universal concept in nonlinear sciences, 2001, p. 8). The oscillators must be independent, however; each one must be able to continue oscillating on its own when the others in the system are absent. Strogatz (Sync: The emerging science of spontaneous order, 2003) concisely described the minimum requirements for synchronization as two coupled oscillators, a feedback loop between them, and a control parameter that speeds up the oscillating process. When the control parameter speeds the oscillating process fast enough, the system exhibits phase lock.
In phase lock, the contributing oscillations all start and end at the same time, with start and end times varying only over a small and rigidly bounded range. If we imagine that the time series of observations produced by a pure oscillator is a sine wave and that its phase-space diagram is a circle, the positions of two or more synchronized oscillators are clustered together as they move around the circle at the same time. Phase synchronization is actually a matter of degree that depends on other matters of degree, such as the tightness or looseness of the coupling produced by the feedback, whether the feedback is unidirectional or bidirectional (or omnidirectional in the case of systems of multiple oscillators), and whether delays in feedback are prominent.
The oscillators in a system are not restricted to pure forms; they can
be forced, aperiodic, or chaotic processes. In fact, three coupled
oscillators are sufficient to produce chaos. This principle has been
exploited as a means for decomposing a potentially chaotic time
series into its contributing oscillatory components. Chaos can
also be controlled by imposing a strong oscillator on the system.
Nervous systems are composed of many oscillating and chaotic
subassemblies; some activate each other while others are inhibitory.
Thus one would anticipate that the products of the nervous system
- movements, autonomic arousal, speech and cognition patterns
- are also fundamentally chaotic, and that pure oscillators are more
often the exception than the rule.
The journal Nonlinear Dynamics, Psychology and Life Sciences
published a special issue on synchronization (April, 2016) that
spanned synchronization within dyadic relationships, relationships
that encompassed teams of four or six people, and finally the Nagent
case (Sulis). The researchers asked questions such as who
synchronizes to whom, in what way, and to what extent? What
conditions affect the synchronization of movements, autonomic
arousal, speech patterns, and brain waves?
Principles of synchronization also lead to some pragmatic questions:
How does synchronization promote more successful therapy
sessions and more effective work teams? Are there conditions
where synchronization is not in the best interest of the dyad or
team? If synchronization, which often occurs at a nonverbal level of
communication contributes to desirable decisions and actions at
a more explicit level, it can also facilitate irrationality, particularly if
stressful conditions are involved.
The idea of social networks was introduced by social psychologists
and sociologists in the early 1950s. Its underlying math comes from
graph theory. In the example diagram, the circles represent people,
and the arrows represent paths of communication, which can be
one-way or two-way. Network graphs are indifferent to the content
of the communication. People interact with each other about all
sorts of things – work, family and other social activities, common
interests, etc. In fact people interact about multiple common interests, so that one graph structure can be superimposed on
More generally, the circles do not need to be people at all. They
can also be airports or other centers in a transportation network,
exchange points in a telephone system, or prey-predator relationships
within an ecological food web. The circles can also represent ideas
that come up frequently in conversations and become connected to
other ideas. Criminologists use network concepts to figure out who
is doing naughty things with whom. Marketing analysts use them to
figure out who is talk about their products and to figure out what
other products that people might like also. Ecologists use the same
constructs to assess the robustness and fragility of an ecosystem
when one of its contributing species is undergoing a sharp population
decline due to human intervention.
Networks can also be analyzed to determine the patterns of
communication to identify efficient and non-efficient alternative
configurations. For instance the star pattern of five people contains
a central hub that communicates with each of the other four
nodes, often bilaterally. The pentagon configuration, in contrast,
depicts five nodes that are communicating with all the other nodes
simultaneously, as in a group discussion.
One type of metric that can be applied to the analysis of networks
is centrality. There are three commonly used types of centrality:
degree, betweenness, and closeness. Degree is the total number of
links that a node can have relative to the total number of links in the
network. Betweenness is the extent to which a node gets in between
any two other connections. Closeness is the extent to which one node
connects to another with the smallest number of links in between.
Closeness is actually the inverse of degree, and people often like to
discussion how made degrees of separation exist between themselves
and somebody else (who might be famous).
As one might surmise, a node can become more central if it has
more links to the other entities in the network. If one were to assess
the frequency distribution of links associated with nodes within a network, the numbers of links are distributed as a 1/fb power law
function, with a small number of nodes having many links and
many nodes having much fewer links. The 1/fb nodes with the most
links become known as hubs in practical application. The pattern
strongly suggests (but does not necessarily prove by itself) that the
network is a result of a self-organizing process.
Studies of network structures, primarily due to Albert Barabasi,
Duncan Watts, and Steven Strogatz, uncovered some interesting
and useful properties of random, egalitarian, and small-world
networks. A random network is just what its name implies: A group
of potential nodes is connected on a random basis. An egalitarian
network is one in which each node communicates to its two next-
door neighbors, but no further. If we were to drop a random
connection into either type of network, the average number of
degrees between any two nodes drops sharply. Hubs start to form,
and we end up with a small-world network in which the average
number of degrees between nodes is approximately 6. Thus, in a
small world, anyone can reach, or be connected to, anyone else in
six links or less; the challenge, however, is to figure out which six
links will do the job.
The robustness of system architectures has numerous practical
implications. If a small world is subject to a random attack, meaning
that the attack is against one node selected at random, the network
will survive because there are enough communication pathways
to link the remaining nodes to each other. If a hub is attacked, the
survival of the network could be in big trouble.
The foregoing dynamics depict how a network could self-organize
into a 1/fb distribution of connections that produce hubs. Hubs
become attractors in the sense that they attract more connections:
people flock to cities, airlines organize their routes around
hub airports, and so on. The avalanche dynamic looms in the
background, however: Physical systems have a limit to their carrying
capacity. Cities become congested and polluted and airports
struggle to maintain flight schedules and proper air traffic control.
One can probably think of more examples. When the carrying capacity is reached, it becomes advantageous to move out of the city, find a new airport to grow, or adapt one's occupation to one that has less competition for resources and attention. The big hub breaks into smaller units that are more equal in size. Per Bak showed, however, that the avalanche produces smaller sand piles that are distributed 1/fb in size. Thus the process is likely to repeat in some fashion.
So far we have focused on the nature of the nodes, but what about the connections? The distinction between strong versus weak ties that has some important dynamical implications. In human communication, strong ties mean rapid dissemination of information within the network. As a result there is a rapid uptake of ideas, which can be convenient many times. The limitation is that the importation of new information becomes unlikely. In those situations, weak ties with other nodes offer the benefit of reaching out to many more nodes, albeit less often, to collect new informational elements.
In a food chain, a predator-prey relationship in which the predator only eats one or a very few specific prey is a strong tie. If an ecological disaster compromises the prey population, the predator is in similar trouble. If the predator has more omnivorous tastes, and thus weak ties to any one particular food source, the predator can leverage itself against a loss of a tasty favorite and survive.
Applications - The Paradigm
There are many applications of nonlinear dynamics in psychology, biomedical sciences, sociology, political science, organizational behavior and management, macro- and micro-economics. We can only provide an overview here and direct our readers to more resources using the links on the panel to the left.
So let's start with the big picture - the paradigm. Nonlinear theory introduces new concepts to psychology for understanding change, new questions that can be asked, and offers new explanations for phenomena. It would be correct to call chaos and complexity theory in psychology a new paradigm in scientific thought generally, and psychological thought specifically. A special issue of Nonlinear Dynamics, Psychology, and Life Sciences
in January, 2007 was devoted to the paradigm question, which actually transcends the various disciplines we study. The highlights of the paradigm are:
1. Events that are apparently random can actually be produced by simple deterministic functions; the challenge is to find the functions.
2. The analysis of variability is at least as important as the analysis of means, which pervades the linear paradigm.
3. There are many types of change that systems can produce, not just one; hence we have all the different modeling concepts that have been described thus far.
4. Contrary to common belief, many types of systems are not simply in equilibrium unless perturbed by some force outside the system; rather stabilities, instability, and other change dynamics are produced by the system as it behaves "normally."
5. Many problems that we would like to solve cannot be traced to single underlying causes; rather they are product of complex system behaviors.
6. Because of the above, we can ask many new forms of research questions and need to develop appropriate research methods for answering those questions, which we are doing of course (see further along on this Resources page).
Nonlinear science is an interdisciplinary adventure. Its growth has been facilitated by the interactions among scientific disciplines, as they are traditionally defined. Scientists soon discover that there are common principles the underlie phenomena that are seemingly unrelated. Consider some quick and blatant examples:
1. The phase shifts that are associated with water turning to ice or vapor follow the same dynamical principles as the transformations made by clinical psychology patients from the time of starting therapy to the time when the benefits of therapy are realized in their lives.
2. The changes in work performance (or error rates) as a person's mental workload becomes too great follows the same dynamics as the buckling of a beam, the materials for which could range from elastic and flexible to rigid and stiff.
3. The growth of a discussion group on the internet parallels that of a population of organisms, which is limited by its birth rate and environmental carrying capacity.
4. The transformation of a work team from a leaderless group into one with primary and secondary leadership roles as its task unfolds bears a close resemblance to the process of speciation in Kauffman's NK[C] model as an organism finds new ecological niches in a rugged landscape. (The former is a less complex version of the latter, however.)
Applications - Psychology
influence. Applications of NDS in psychology include neuroscience; psychophysics, sensation, perception and cognition; motivation and emotion, group dynamics, leadership, and collective intelligence; developmental, abnormal psychology and psychotherapy; and organizational behavior and social networks. The Society's book project, Chaos and Complexity in Psychology: The Theory of Nonlinear Dynamical Systems provides a state-of-the-science compendium (through 2008) of psychological research on the foregoing topics in textbook format. The chapter authors make frequent contrasts between the conventional scientific paradigm and the nonlinear paradigm.
The article that follows in the list of links, "Chaos as a Psychological Construct" examines the concept of chaos as it has appeared in a wide range of psychological literature. Uses of the construct range from common use of the word chaos, which usually has no intended reference to formal nonlinear dynamics, to applications where chaos is meant seriously. The research efforts that follow on this page, in psychology and elsewhere, use NDS constructs literally and analytically.
The resources list for psychology includes special issues of Nonlinear Dynamics, Psychology, and Life Sciences that are focused on psychomotor coordination and control, creativity, brain connectivity, developmental psychopathology, organizational behavior, and education, and interpersonal synchronization. Other special issues are in the works, and a nnouncements of new projects will appear on our NEWS page and the NDPLS home page.
To make matters more interesting, many areas of psychology have embarked on the "biopsychosocial explanation of everything." The meaning of the remark is that there is a growing awareness of the interconnections among brain and nervous system activity, cognitive processes and social processes now that we are learning much more about all three realms. The opportunities for new works - and stronger explanations for phenomena - in nonlinear dynamics are extensive. Consider what else is going in the biomedical sciences (next).
Applications - Biomedical Sciences
Some of the first applications in the biomedical sciences explored the comparisons of the fractal dimensions of healthy and unhealthy hearts, lungs, and other organs. Larger dimensions, which indicate greater complexity, were observed for the healthy specimens. This finding gave rise to the concept of a complex adaptive system in other living and social systems. In the area of cognitive neuroscience, memory is now viewed as a distributed process that involves many relatively small groupings of neurons, and that the temporal patterns of neuron firing contain a substantial amount of information about memory storage processing. The temporal dynamics of memory experiments can elucidate how the response to one experimental trial would impact on subsequent responses and provide information on the cue encoding, retrieval, and decision processes. One might examine behavioral response times and rates, the transfer of local electroencephalogram (EEG) field potentials, similar local transfers in functional magnetic resonance images. In light of the complex relationships that must exist in processes that are driven by both bottom-up and top-down dynamics, the meso-level neuronal circuitry has become a new focus of attention from the perspective of nonlinear dynamics.
Dynamical diseases, which were first identified by Leon Glass, are those in which the symptoms come and go on an irregular basis. As such, the underlying disorder can be difficult to identify and the symptoms can be difficult to control unless one reframes the problem as one arising from the behavior of a complex adaptive system. This notion has now carried over to the analysis of psychopathologies with some success. There are, in turn, further implications for medical practice, David Katerndahl and other writers have observed that the mainstay of medical practice in most countries still revolves around the single cause mentality, and reorganization of the service delivery system around complex adaptive systems could be necessary to produce effective and efficient care.
Applications - Organizational Behavior
Organizational Behavior grew up from a combination of influences, most of which stem from Industrial/Organizational Psychology, although there are important influences from other areas of psychology, sociology, cultural anthropology, and economics. In short, it is the study of people at work. The subject area had undergone a few paradigm shifts over the last century while trying to answer just one question, "What is an organization?"
1. During the rise of large corporations in the late 19th century there was no answer. Business organizations were managed on an ad hoc basis, which is to say the opportunities for chaos and confusion in the conventional sense abounded. Organizational leaders had few role models other than the military and the Catholic Church, both of which were command-and-control structures then and not much different now. This is not to say that individuals did not have more productive views and intuitions about human nature.
2. Weber, a sociologist introduced the concept of bureaucracy circa 1915, which was intended to instill rationality and efficiency where there was none before. People were separate from their roles. It was roles that had rights and responsibilities. The drive toward impersonality and supposed efficiency had a negative consequence, which was to produce impersonal and mechanistic enterprises with people being plug and play machine parts.
3. The nature of organizations changed drastically with the introduction of humanistic psychology, which extended not only to the understanding of individual personality, but the role of people in organizations. The full capabilities of an organization could be unleashed by giving normal human tendencies to grow and achieve opportunity for expression. Here entered psychologist Kurt Lewin, whose platform work on organizational change (a.k.a. organizational development) facilitated the first understanding of social change and intervention processes. Most of the early change efforts were directed at changing an organization from a mechanistic and authoritarian bureaucracy to a humanistic enterprise.
4. The transition to the organic model was not as dramatic as the previous change in viewpoints. The new understand was, nonetheless, that the organization itself is a living entity, and not simply humans in a nonlinear shell. Organizational life bore many similarities to the biological life of single organisms. The organic model was an important step along the way to what came next.
5. The current paradigm, and answer to the question, "What is an organization" is: A complex adaptive system. Here we can observe all the principles of nonlinear dynamics - chaos, catastrophe, self-organization and more - occurring in relationships between people and their work, the functionalities of work groups, work group dynamics, and organizational change. This line of thinking has broad implications for leadership before, during, and after an organizational change process. In fact, change is now understood to be ongoing and business as usual, and not confined to deliberate interventions.
The new paradigm is explicated in "A Complex Adaptive Systems Model of Organizational Change," by K. J. Dooley. The article debuted in the inaugural issue of NDPLS in 1997, and remains a landmark in our new understanding of organizational behavior.
Organizational behavior, or I/O Psychology, was among the first application areas of NDS in psychology, with the first contributions dating to the early 1980s. Extensive work on organizational theory has followed since then.
Human factors (engineering) started as the psychology of person-machine interaction - everything that took place at the interface between person and machine. There was also some concern with the interface between the person and the immediate physical environment. The psychological part of the subject matter is primarily cognitive in nature. The area has merged in scope with ergonomics, which started with interaction between people and their physical environment and numerous issues in biomechanics.
In light of new technological transitions and the growth in connected networks of people and machines, this area of study has expanded further to include group dynamics regularly, and organization-wide behavior in some of the most recent efforts. Of particular importance here, there is now a further recognition of systems changing over time and the nonlinear dynamics involved, such that it is now possible to define NDS as a paradigm of human factors and ergonomics. Applications of particular interest include nonlinear psychophysics where the stimulus source and observer are in motion, control theory, cognitive workload and fatigue, biomechanics connected to possible back injuries, occupational accidents, resilience engineering, and work team coordination and synchronization.
Applications - Economics & Policy Sciences
The relationship between economic concepts and those from physics, nonlinear and otherwise, dates back to the early 20th century. The Walrasian equilibrium and the Nash equilibrium decades later are two notable examples. Game theory became a key concept in psychological, economic, and other agent-based dynamics since that time. The advent of evolutionary game theory in the mid-1980s extended the reach of nonlinear simulation studies. Rosser's 3-volume set (Menu 3) compiles the critical work in nonlinear economics that appeared in many journals through 2004. Even more has happened since then. Finance is one of four broad areas of economics that is replete with nonlinear dynamics. Macroeconomics is another challenging dynamical domain. Some schools of economic thought attempt to reduce system-level events to the decisions of individual financial agents. NDS studies in macroeconomics, however, are more consistent with other schools of economic thought that assume that one cannot assume a particular system-level outcomes, such as inflation or unemployment from knowledge of forces acting on individual economic agents. Ecology is another active area of nonlinear science. The discovery of the logistic map function actually arose in the context of population dynamics, and birth rates and environmental carrying capability change.
Ecological economics is a third broad area that lends itself to NDS analysis. Topics in this group include environmental resource protection and utilization, agricultural management, and land use and the fractal growth of urban areas. Cellular automata have been useful tools for studying urban expansion.
Evolutionary economics is the study of behavior change on the part of microeconomic agents, institutions, or macroeconomic structures. Here one finds chaos and elementary dynamics, game-theoretical experiments with evolutionarily stable states, and multi-agent simulations based on genetic algorithms or related computational strategies. An important part of policy making in any of these area is the ability of the decision makers to recognize the "time signatures" of various dynamical processes and plan accordingly. The efficacy of the current wave of institutional policymakers for doing so is a different question entirely.
Applications - Education
Applications in education cover a range of individual-level, group-level, and institurional level topics. Examples include student learning and motivation dynamics, social dynamics in the classroom or school, student-teacher interactions, career development, absenteeism, and the long-run effect of policy concepts on the success of metropolitan school systems. There is some parallel between the topics addressed as education issues and those in organizational behavior more generally.