jueves, 13 de noviembre de 2008

La construcción social de la realidad.

La suma de instituciones forma las sociedades. Toda actividad humana esta sujeta a habituación en un proceso: problema-actividad-solución-habito-tipo-Institución.

La facultad de habituación esta preprogramada genéticamente, las habituaciones se revisten de un carácter significativo (simbólico). La habituación remplaza al instinto y da seguridad psicológica al sujeto.

Típificar es caracterizar, ajustar algo a un molde. La tipificación instituye la historia.
La tipificación implica que la acción X se realice por el actor Y.

Institución: tipificación-habituación-selección-sistema de control social-practica institucionalizada. Cuando esto se cumple aparece la tradición.
Cuando esto no se cumple aparece la ley. Cuando la institución se violada repetidamente parece la sanción (típica con una pena). No hay pena sin sanción, no hay sanción sin ley. Las instituciones nacen de situaciones cara a cara, donde cada actor cumple un rol. Hay roles separados y roles en común. La situación cara a cara es una interacción de actores.

Actores:
A------------B
A atribuye motivos a B
B atribuye motivos a A
Repeticion
Tipificacion:
A tipifica rol a B
B tipifica a rol A
Aparecimiento del rol
Aparece la división del trabajo.
Aparece la estratificación social.

El tiempo y la repetición permiten que la acción del otro sea previsible.
La institución aparece cuando A y B, transmiten su rol a C y la institución se objetivisa, se cosifica y se inflexibiliza. La institución toma una realidad propia independiente, se auto produce y se vuelve coercitiva para el otro.

A + compartida la tradición + cosificación: ethos-mores-axiomas-ley.
Este entretejido social es la realidad social, la red significante, el mapa y el territorio cultural que los hijos heredan, en un proceso de operaciones concretas en sincronía.

El mundo objetivo-institucionalizado y controlador de sus integrantes
Así la institución es acción social objetivada donde el producto actúa sobre su causa en circularidad: La sociedad es un producto humano y el sujeto es un producto social.

La realidad social, mientras + masiva + real.
Cuando el yo internaliza el control sobre si mismo aparecen los valores, que son agentes internalizadores del control social.

Todo el proceso escapa ala memoria, por eso se justifica. La sociedad se justifica a si misma por medio de la autojustificación de las instituciones así la Legitimación es el proceso de la autojustificación de las instituciones.

El entendimiento humano le superpone “lógica” a todo el proceso y justifica
(Legitima) y así dota de significado a el orden social (auto controlado). Esta lógica es auto explicada y las instituciones se integran de facto en la autoexplicación social,
Por la necesidad de otorgar sentido a esta organización autosostenida.
La lógica de lo social, es gestor, mente, verdugo y guardián del orden social.
Los significados objetivados y compartidos en la sociedad se sedimentan y este patrimonio cultural se transmite através del lenguaje, esta transmisión es una reinterpretación mediada por la conciencia en forma de conocimiento sujeto a un control social externo e internalizado.

Este proceso carece de coherencia, si existiera coherencia entre la institución y la justificación no fueran necesarios los dispositivos de de control social.
En un proceso donde el yo su alter ego el rol, la persona y la personalidad se vuelven agentes de autocontrol social.

Así la actividad humana se tipifica en roles, roles que sirven para legitimar a otros roles, la división del trabajo es división de roles y la estratificación social es estratificación de roles, sin conciencia de roles no hay instituciones ni sociedad.

La sociedad es un sistema de instituciones.
La institución es un sistema de roles.
La sociedad es un, macrosistema de roles
La cultura es un sistema de significados y significantes culturales, esto es los universos simbólicos.

Dentro de la estatificación social aparece la necesidad de profesionalizar la justificación, aparecen los contadores profesionales de mitos. El mito es teoría pura, se ritualiza y se sistematiza y aparecen las religiones.

Toda cohesión social es producto de una autojustificación de roles. La estratificación produce autonomía de roles estos se estratifican en instituciones civiles, religiosas, científicas y hasta sociedades secretas. El extremo de este proceso es considerar a los productos sociales como cosas esto es la reificación.

La reificación de las cosas naturales deriva en el derecho natural, la reificación de los mitos religiosos deriva en el derecho divino. Los fenómenos sociales se toman por cosas, los procesos y productos humanos se toman por verdades y se establecen los valores.

Los sistemas teóricos complejos terminan siendo deificados, la reificación es un proceso psicológico, donde los roles se deifican para poder entenderlos y justificarlos: La majestad del poder.

La legitimación es la autojustificación de los roles para autojustificar las instituciones y por ende ala sociedad.


4 niveles de legitimación.
1. incipiente
2. proposicional.
3. teórico (ideología)
4. Dogma. La justificación final es la teoría pura transmutada a dogma.

Los sistemas sociales están en evolución continua desde el nivel incipiente hasta el dogmático. El sistema tiende a perpetuarse (conservarse) y perpetuar a sus agentes de control (coerción-coacción) y esto produce el terror social. El ajuste social se da por retroalimentación. Hay un proceso de mantenimiento que prepara y redirige para proteger y defender.

Una función del dispositivo de control es la resocialización de los miembros desviados, aparece la terapéutica. La terapia busca la re-viación social, anulando la individualidad y potenciando la sociabilidad. (Homo duplex). El sistema social condena a quienes no cumplen sus estándares, los casos de marginación son casos de segregación


El poder social produce realidad social. Los significados sociales van evolucionando hacia el dogma junto a ellos evoluciona la conducta represora de los agentes de control.
Los lideres son guardianes de la institución, las instituciones forman una estructura unificada de roles que monopolizan la realidad del sistema y la estructura tiende ala morfoestasis (conservación). Así la ideología es alienante por que esta al servicio del poder definiendo ala realidad según el interés concreto del amo del sistema, generando cohesión y servicio a ese amo por el mecanismo de identificación.

Así los universos simbólicos se auto justifican, auto explican y auto controlan.

El hecho social objetivado pasa a tener un significado subjetivo para el sujeto, esta facultad de subjetivar lo objetivado por el otro, hace al sujeto, sujeto de su cultura y habitante de su mundo social.

Al internalizar el mundo social, lo hacemos significativo y lo mediamos, el sujeto de la cultura es un biosistema perturbable y perturbador de otros biosistemas.

El otro media y filtra el mundo para mí, por este proceso el sujeto se estructura en ser social en un proceso de formatización. El sujeto va a responder a su núcleo primario de socialización en la medida que su núcleo responda en su estructuración en una doble implicación de gestor-gestado.

En este proceso se consolida la identificación que auto verifica el discurso del otro en marcando esa identidad en parámetros recursivos ideosincráticos, bucles de retroalimentación, así la identidad es tautológica.

Solo cuando el sujeto logra coincientizarse como parte de una realidad mayor que excede su núcleo socializador primario, aparece el otro generalizado y muta en el sujeto de una sociedad global, esta listo para la socialización secundaria que a estas alturas ya esta en proceso, sin perder nunca el vinculo con esa socialización primaria que le sirve de vinculo con el mundo.

Los agentes de la socialización primaria son los familiares y los agentes d socialización secundaria son agentes institucionalizados, encarnan roles institucionalizados, representan instituciones y transmiten estos roles a los neófitos.

El nuevo rol trae nuevos códigos e ideología propios de su institución. El sujeto logra encarnar el rol cuando logra la maestría del lenguaje del rol. El rol trae un universo de símbolos, pero el mundo de la socialización primaria no desaparece.

El paso de la socialización primaria a la secundaria suele estar marcado por rituales sociales cognitivo afectivos. Los conclaves socializadores, son conclaves alienadores. Cuando los valores de la socialización primaria chocan con los nuevos valores de la socialización secundaria el sujeto suele entrar en crisis. Si el mundo social a de remplazar al mundo familiar este proceso debe ser devastador. El sujeto (alienado) puede llegar al autosacrificio cuando siente que se lo exigen los valores sociales (ideológicos) del rol que ha encarnado.

Para el sujeto social, sin el otro no hay realidad, por que la realidad es realidad social, cuando el núcleo del sujeto se desintegra o muta el sujeto se altera.

La tradición se mantiene en diálogos cara a cara, cuando el dialogo se rompe la tradición cae en desuso y la realidad cae en crisis, los conclaves científicos y los rituales religiosos cumplen la función de prevenir las crisis de realidad.

La coerción social, logra que el sujeto se mantenga sujetado al dialogo social. Si se rompe la coerción y un sujeto se desvía en conjunción con otros puede lograr formar un submundo social, se segrega y forma su propio mundo con sus propios códigos y valores que se mantienen en cuanto el grupo se auto segrega del exo-sistema. Y este universo social entra en la típica evolución en pro de la institucionalización que derivara tarde o temprano en el dogma, generando su propia ideología, mitología y teología.

Todo el proceso anterior derivara en la identidad individual del sujeto en relación a su mundo social pero la identidad nunca es colectiva.



Fuente: la construcción social de la realidad
Berger y Luckman.







Eduardo Roldós Arosemena.

martes, 11 de noviembre de 2008

COMPLEX SYSTEM

http://el.www.media.mit.edu/groups/el/projects/emergence/

A complex system is a system for which it is difficult, if not impossible to restrict its description to a limited number of parameters or characterizing variables without losing its essential global functional properties.
This definition is proposed with reference to our experience with the study of socio-technical cooperative systems
Notes
A more precise definition of a complex system: Formally, a system starts to have complex behaviors (non-predictability and emergence etc.) the moment it consists of parts interacting in a non-linear fashion. It is thus appropriate to differentiate between a complicated system (such as a plane or computer) and a complex system (such as ecological or economic systems). The former are composed of many functionally distinct parts but are in fact predictable, whereas the latter interact non-linearly with their environment and their components have properties of self-organization which make them non-predictable beyond a certain temporal window.

A truly complex system would be completely irreducible. This means that it would be impossible to derive a model from this system (i.e. a representation simpler than reality) without losing all its relevant properties. However, in reality different levels of complexity obviously exist. If we are interested in situations which are highly structured and governed by stable laws, then it is possible, without loosing too many of the system’s properties, to represent and model the system by simplification. Thus, the essential question is to know to what extent the properties of the socio-technical systems that we analyze and design fall into one or the other of these situations. In other words, to what extent can we make an abstraction of microscopic interactions in order to understand macroscopic behaviors? In what measure are microscopic interactions linked in a non-reducible way with the laws that govern more structured behaviors? Finally, is it possible to explain the most structured behavior using rules which control the microscopic behavior (the principle of emergence)? This last question is important from an epistemological and methodological point of view: if we consider theoretical economy, it can be preferable to generate the structural property of a system using knowledge of its microscopic properties (emergence), rather than suggest its macroscopic properties and only validate them with an analytical process.

The reduction of complexity is an essential stage in the traditional scientific and experimental methodology (also known as analytic). After reducing the number of variables (deemed most relevant), this approach allows systems to be studied in a controlled way, i.e. with the necessary replication of results. This approach in itself need not be questioned. However, when considering complex socio-technical systems it is appropriate to analyze precisely the limits of the approach.

Some Properties of Complex Systems:

Non-determinism and non-tractability. A complex system is fundamentally non-deterministic & non-tractable.
Limited functional decomposability.
Distributed nature of information and representation.
Emergence and Self-organization.

Property 1: non-determinism and non-tractability. A complex system is fundamentally non-deterministic. It is impossible to anticipate precisely the behavior of such systems even if we completely know the function of its constituents.

Property 2: limited functional decomposability. A complex system has a dynamic structure. It is therefore difficult, if not impossible, to study its properties by decomposing it into functionally stable parts. Its permanent interaction with its environment and its properties of self-organization allow it to functionally restructure itself.

Property 3: distributed nature of information and representation. A complex system possesses properties comparable to distributed systems (in the connectionist sense), i.e. some of its functions cannot be precisely localized. In addition, the relationships that exist within the elements of a complex system are short-range, non-linear and contain feedback loops (both positive and negative).

Property 4: emergence and self-organization. A complex system comprises emergent properties which are not directly accessible (identifiable or anticipatory) from an understanding of its

Non-determinism and non-tractability

A complex system is usually non-deterministic & non-tractable
In this example, the Medic (Med) is telephoning an external agent (C).

Due to the proximity relationship between the medic and all other agents, the conversation is opportunistically listened to by the agent O who then sends an ambulance (because she inferred the case discussed by the medic was urgent)

Non-determinism of socio-cognitive processes is often considered as being due, either to a lack of knowledge of the observer about the analyzed system, or to a disturbance of the system as a result of unforeseen causes (e.g. exterior events or noise etc.). An analysis of the properties of complex socio-technical systems suggests that non-determinism can have an important functional role. We consider one of the most usual mechanisms concerning cooperative systems: broadcasting. We will show that this mechanism is non-traceable (i.e. that it is difficult, if not impossible, to describe explicitly the information flows that are relevant in understanding how a collective functions) and that it provides a structure for the management of the memory of the collective.

An example of the broadcasting mechanism. A caller, C, telephone a medic (Med) at the emergency centre to request an ambulance. This communication can be overheard by several people depending on their geographical position and the volume of the communication. These people can be either authorized unauthorized, interested or disinterested interlocutors. The fluctuating status of the interlocutors, as well as their geographical positioning or their level of involvement with a task, will significantly influence the development of the common knowledge of the collective. In this example, we can see (in 3) that agent O overheard the conversation between the caller and the medic (1 and 2) because of his spatial proximity to the doctor and the volume of the communication. As a result, agent O dispatched an ambulance without the medic making an explicit request.
The cognitive dimensions of broadcasting are varied (audio, visual, gesture, etc.) and each one contributes to making the process non-deterministic. Some of the main factors contributing to this mechanism are: the number of people present at the time of the communication act, their status (authorized or unauthorized interested, etc.), their availability and the context etc.
It is extremely difficult to trace the flow of associated with this type of communication. Neither the actors involved, nor the observer have the means or the cognitive resources to know who heard the message and even less to know how it was interpreted. In addition, it is often very difficult to separate the environmental factors from the internal factors.

Limited functional decomposability

Plasticity in the division of Labor in Social Insects
Different activities are often performed simultaneously by specialized individuals, but the division is rarely rigid.

Workers switch tasks to adjust to changing conditions maintaining the colony’s variability and reproductive success.

Factors which cause the change in role are due to internal colony perturbations or external challenges, e.g.: food availability, predation, climate change.

This property of complex systems is difficult to understand intuitively since it goes against the principles of the dominant functionalist culture. According to the traditional analytical approach, a system that is functionally decomposable is a system whose global functioning can be completely deduced from knowledge of the function of its sub-components. To take a trivial example, if we know the function of each element of a car (brakes, distributor, engine etc.) it is possible to calculate the global function of the vehicle by combining the functions of each element. Systems theory (cybernetic, automatic) is one of the disciplines essentially dedicated to formalizing this approach.
A truly complex system cannot be represented by combining together a collection of well defined functional components. A principal obstacle to the functional decomposability of complex systems is the dynamic and fluctuating character of its constituent functions. The interaction with the environment, as well as the learning and self organization mechanisms makes it unrealistic to regard such systems as structurally stable.

The notion of distributed information is largely polysemic, conveying widely different concepts. In its most commonly accepted meaning, a system is said to be distributed when its resources are physically or virtually distributed on various sites. Thus, a machine (a computer for example) can distribute its calculations amongst several remote sites and assemble the results according to a pre-defined algorithm. Equally, an operator can distribute his or her work tasks and tools according to a particular strategy. The concept of distribution supports the concept of redundancy, when some distributed resources are redundant.
The notion of distributed representation also exists in the field of cognitive psychology. It covers the fact that, in the interaction between an actor and his environment, artefacts (tools) play an important functional role in the organization of the reasoning and the transmission of knowledge. To illustrate this principle, we will take the frequently used example of paper strips in the domain of air traffic control. Paper strips are small pieces of paper on which aircraft characteristics, such as its call sign, its destination and its route, are written. It has been shown that these strips help the controllers to represent information to themselves (for example by having the strips organized on the strip board according to the dynamic properties of the planes) and also to cooperate between themselves. Thus, we can speak about distributed representation, since some cognitive properties (such as memorizing and structuring of the problem etc.) are partially supported by artefacts in the environment. In one way, this notion is close to the concept of physically distributed systems.

Finally, we could introduce a third meaning to the notion of distributed systems which stems from connectionist models and conveys essential concepts for understanding the robustness of the collective in processing data. In the connectionist meaning, a distributed system is one where it is not possible to localize physically the information since it is more or less uniformly distributed between all of the objects (or actors) in the system.

We can see that the term “distributed representation” is inappropriate here since it is impossible to identify any form of representation in such a network. The representation is “dissolved” either in the nodes of the system or in the links. Thus, a distributed system, in the connectionist sense, does distinguish between concept, representation, and context, since these three entities are “encoded” simultaneously on the same support (nodes and links).

We can say that a truly cooperative system works on both representational and connectionist modes. This is why the system is particularly robust in complex environments, which are unpredictable and non-deterministic.

The following example is concerned with our study on the reorganization of the emergency centre. The aim of the collective in the centre is to maximize cooperative behavior between the actors, in order to respond in the best possible way to events in the environment (such as unexpected calls, work peaks, changes in the physical position of the agents, etc.). The efficiency of this type of collective is based on a situation of co-presence which allows information to be distributed by broadcasting and “floating ear”.

In the case of a normal workload, it is the proximity between the agents which allows them to keep informed of what is said in the collective (floating ear) and to regulate locally the efficiency of information distribution (by talking more or less loudly, by adjusting the volume of the loud speaker and by adopting adaptive ostensive behaviours). In addition to the information distribution between agents there is there is important interaction between the environmental factors (e.g. noise level and space constraints within the room) and more central processes (such as the control of the modes of communication). When a call, which relates to a previous call, is taken by another agent (i.e. one that did not take the initial call), the system is robust enough overall to be able to redirect the call to the correct agent.

Such a system can be regarded as complex because part of its functions (here the functions of information sharing and information distribution) cannot be reduced to a representation where it is possible to locate precisely a relevant piece of information. Neither the actors nor the observer can, at a given moment, give a deterministic plan of this process. Moreover, as we saw previously, the structural properties of the communication system are under local control: each agent can control the way in which he locally distributes the information. Understanding how such a system works requires having a model of this type of dynamics, including mechanisms of training, of self regulation and control of the interaction with the environment.
References:

Cilliers, P. , 1999, Complexity & Post modernism

COMPLEX SYSTEMS, EMERGENCE AND SELF-ORGANIZATION

http://el.www.media.mit.edu/groups/el/projects/emergence/

Emergence is the process of deriving some new and coherent structures, patterns and properties in a complex system.

Emergent phenomena occur due to the pattern of interactions between the elements of the system over time.

Emergent phenomena are observable at a macro-level, even though they are generated by micro-level elements.

First and Second Order Emergence.

Emergence is one of the characteristics of a complex system and has become such a important idea in complex systems that perhaps it deserves a special mention.
Emergence is the process of deriving some new and coherent structures, patterns and properties in a complex system. Emergent phenomena occur due to the pattern of interactions (non-linear and distributed) between the elements of the system over time. One of the main points about emergent phenomena is that they are observable at a macro-level, even though they are generated by micro-level elements.
In terms of social organizations, emergent behavior is an important concept. For example, we could consider the occurrence or social norms within group emergent phenomena. Note that from a modeling point of view the identification of some behavior as being emergent depends of what has been modeled. To continue the example of social norms above, social norms *could* have been one of the elements modeled in a system. Only in the case where they were not modeled explicitly, could they be considered emergent phenomena.

Nigel Gilbert (editor of JASSS) makes an interesting comment about emergent behavior in human social organizations compared to non-human social organizations (such as a collection of ants). The differentiation between the two organizations lies in the ability to reason. Specifically, ‘people have the ability to recognize, reason about and react to human institutions, that is, to emergent features. Behavior which takes into account such emergent features might be called second-order emergence’ (as opposed to first-order emergent behavior). This has implications for modeling human social situations since we might need to model what *effect* the macro-level properties have on agents actions.

History of Complex Systems

Henri Poincaré showed the new conceptual difficulty: how a completely causal system could have indeterminate behavior.

Non-linear systems & neural networks

Chaos

Distributed & self-organised systems

Artificial Life and Agent-based Societies

The theories of complex systems have been developed along three complementary, but nevertheless distinct, axes: the theory of non linear systems, the neural network approach and the theory of distributed or self organized systems.

Historically, the notion of complex systems was born at the beginning of the century when H. Poincaré worked on the equations used to predict the trajectory of planets. H. Poincaré showed that it was mathematically impossible to find an exact solution to these equations even for a system as simple as that containing three planets interacting in a non-linear fashion.

Poincaré revealed to the scientific community a new conceptual difficulty: even a completely causal system (a system where the behavioral rules are perfectly known) could have indeterminate behavior. Put another way, he showed how a simple system can explode into complex and unpredictable behavior.

The research soon faced many obstacles: one of them in the form of cultural difficulties and H. Poincaré himself reassessed the epistemological consequence of his work on non-linear systems. The point that would have intrigued H. Poincaré the most is how a perfectly determinable system, from a functional point of view, could have non-predictable behavior. Another difficulty came from the lack of computing power which would have been needed to find approximate solutions to problems that do not have exact solutions and thus explore the new field of complex systems.

Nevertheless, the school of non-linear systems brought many new conceptual and methodological insights. These contributions were not directly applicable to the study of socio-technical systems which are the systems of interest to ergonomists, designers and sociologists.

Non linear systems properties were also investigated through the development of neural networks. Neural networks researches emerge during the 50s with the perceptron. They were used in order to mimic the behavior of real neurons and to explore their classification capabilities. Due to their non linear properties, these systems have very interesting properties of classification and extrapolation which has been used as a metaphor for cognitive processes.

Later, the study of distributed and self organized systems overcame this difficulty and provided new perspectives in modeling social and cognitive systems.

Basically, the theory of distributed and self organized systems is based on the fact that a population of independent and autonomous agents interacting only locally may produce “intelligent” global behavior. The system is then said to have properties of self organization.

This approach has a long lineage beginning with the study of connectionist systems to artificial life and agent based societies. The methodological and philosophical roots of the distributed and self organized systems are drastically different from the classical analytical approaches and mainly due to the fact this paradigm does not use the concepts of representation.

The distributed and self organized approach found many applications in fields ranging from the study of micro societies (ethology) to the study of human organizations and anthropology.

Henri Poincaré, a mathematical physicist, attempted to answer the question of whether the solar system was stable forever, or if some planets would just simply drift off. This required an attempt to solve the celestial 3-body problem.
The 3 Body problem: Given 3 bodies (e.g. Sun, moon, Earth) and their initial positions and velocities, the problem is to determine the motion of the 3 bodies attracting one another according to Newtons law of gravity. Whilst the it sounds quite straightforward, the problem is surprisingly difficult to solve.
Issac Newton had solved the 2-Body problem and a solution was sought for the 3-Body problem and more generally the N-Body problem).
Given the deterministic way of thought, people believed that they could predict into the future provided they have sufficient information. Thus, given sufficient information they could easily solve the 3-Body problem.
In 1887 the King of Sweden and Norway, Oscar II, initiated a mathematical competition to celebrate his 60th Birthday in 1889. Henri Poincaré selected the 3-Body problem (actually, he considered a 9-Body problem: the then known about 8 planets plus the Sun. However, he realized that the minor components of the solar system would produce perturbations on the planets and thus the problem was closer to a 50-Body problem. He immediately saw the difficulty with this and restricted himself to the 3-Body problem.)
Poincaré was familiar with the then current algebraic techniques and their limitations. However, he started to look at the problem from a different point of view and decided to try a geometric approach. This approach was ground-breaking and although he had failed in solving the problem, he was awarded the prize.
His revolutionary work was to be published in Acta Mathematica. However, during the publication process, Edvard Phragmen (a Swedish Mathematician) noticed a serious error in Poincarés work. The editor of Acta Mathematica immediately stopped publication and asked Poincaré to review his work. With further effort, Poiincaré looked again at his data (primarily patterns on the slices in phase space) and realised that the orbit of a planet in a case such as his could not be calculated far into the future. He was shocked by the results and rewrote his paper.

After the final publication, Poincaré abandoned the 3-Body problem, although the strange results he had obtained bothered him and he made constant referrals to them throughout his lifetime. Given the lack of interest from the scientific community and the advent of computers to analyze efficiency the work, the 3-Body problem and it’s bizarre implications went out of vogue. It wasn’t until much later that the scientific community realized that Poincaré had predicted chaotic motion and broke the ground in the new field of chaos.



In order to explain how non linear systems are related to complexity, we will consider a very simple example of a non linear world: two concurrent populations of mice and cats living in a closed environment. Cats eat mice, but if they are not enough mice, cats start to starve and their population decreases allowing the mouse population to increase again. How can we simply model such an equilibrium?

In the early 1970s a mathematician, R. May, studied an equation which could approximately represent such interaction. This well known equation is called logistic equation :

Xn+1 = kXn - kX2n = kXn (1-Xn)

It says that the population of mice at year n+1 (Xn+1) is subject to two opposing trends : a growing factor (k) due to their breeding rate and a decreasing factor (-kX2n) which says that the mice population cannot grow too much because cats eat them. Note that, at this time we are not interested if this equation mimics exactly the interaction but only to its dynamic properties. We could first notice that this equation is non linear due to the corrective factor (-kXn2). Non linear means that if you double the input (Xn), you will not double the output.

Now we have our model, we can see it is perfectly deterministic (that means if we know the initial population and its growing factor k, we can always compute how the population will evolve with time. To do so, we can choose to use a time diagram, such as that shown in the leftmost figure at the top of the slide, and we will see that the mice and cats population oscillate approximately in synchrony which can easily be understood: when mice are seldom, the number of cats will decrease due to a lack of their preferred food and mice will take this opportunity to breed again, and so on. Alternatively, we can represent the same phenomena on a phase diagram which is an equivalent type of representation (the rightmost figure at the top of the slide).

Now we can analyze what happens when there is a change in the value of the mice breeding factor k.

If k is small, e.g. k=1.2 (mice do not reproduce very fast), the mice population will stabilize over the following years (Fig 2). If the growing factor increases a little bit, (k=1.5), the system behaves gently: the mice population increases in consequence. However, when k = 2.3 something new happens, Fig 2 shows that the mice population starts to oscillate between two values (oscillation of period 2); for k = 2.5, the oscillation is of period 4 which means that it takes four years for the population to come back to the same value. Finally, at k = 3, the process is no longer periodic. The mice population jumps incessantly among an infinite number of values in a way which is deterministic but cannot be predicted over a long period of time.

The bottom four diagrams show the transition from order to chaos for two populations interacting in a non linear way. Here, the mice population has been reported in relationship to its growing factor k. It is possible to see that when k=3, the mice population changes in an erratic (chaotic) way over the years.

From this simple example, we can already see some interesting results:
1) Chaotic behaviour can arise even in a very simple system. In our case, the two populations where related to a simple non linear equation which is fully deterministic.

2) Complexity can arise only from two facts: iteration (feedback from one year to the other) and non linearity in the feedback mechanism. Then, it is not necessary to have many interacting systems in order to get complexity.

3) Even a fully deterministic system (the mouse population at year N is fully specified if we know it at year N-1) can show chaotic behavior which means unpredictability over a certain period of time.

4) Deterministic behavior can be seen as a special case of chaotic behavior. If we take k=2.85 we can observe a small window of stability. If the mice-cat population has this growing factor value in this window, its behavior will be perfectly deterministic. It will be possible a find out rules or equations that allow a perfect computation of mice population over the time. This phenomenon is called intermittency (a period of order in a universe of randomness).
This characteristic behavior raises interesting questions such as: to what extent are an ordered system and its chaotic version both faces of one indivisible process? Is our familiar rule based world just an island of intermittency in the midst of chaotic universe?

References.

The N-Body Problem:
http://members.fortunecity.com/kokhuitan/nbody.html

Chaos and Henri Poincaré:
http://zebu.uoregon.edu/~js/21st_century_science/readings/Parker_Chap3.html
The 3 body problem
http://astro.u-strasbg.fr/~koppen/body/ThreeBody.html