State-Space Trajectory and Basin of Attraction

A system that undergoes unexpected and/or violent upheaval is always attributed to as being facing up to a rupture, the kind of which is comprehensible by analyzing the subsystems that go on to make the systemic whole. Although, it could prove to be quite an important tool analysis, it seldom faces aporetic situations, because of unthought behavior exhibited. This behavior emanates from localized zones, and therefore send in lines of rupture that digresses the whole system from being comprehended successfully. To overcome this predicament, one must legitimize the existence of what Bak and Chen refer to as autopoietic or self-organizing criticality.

…composite systems naturally evolve to a critical state in which a minor event starts a chain reaction that can affect any number of elements in the system. Although composite systems produce more minor events than catastrophes, chain reaction of all sizes are an integral part of the dynamics. According to the theory, the mechanism that leads to minor events is the same one that leads to major events. Furthermore, composite systems never reach equilibrium but evolve from one meta-stable state to the next…self-organized criticality is a holistic theory: the global features such as the relative number of large and small events, do not depend on the microscopic mechanisms. Consequently global features of the system cannot be understood by analyzing the parts separately. To our knowledge, self-organized criticality is the only model or mathematical description that has led to a holistic theory for dynamic systems.

The acceptance of this criticality as existing has an affirmative impact on the autopoietic system in moving towards the point aptly called the critical point, which is loaded with a plethora of effects for a single event. This multitude is achieved through state-space descriptions or diagrams, with their uncanny characteristics of showing up different dimensions for different and independent variables. Diagrammatically, a state-space or Wuensche Diagram is,

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and

 

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In all complex systems simulations at each moment the state of the system is described by a set of variables. As the system is updated over time these variables undergo changes that are influenced by the previous state of the entire system. System dynamics can be viewed as tabular data depicting the changes in variables over time. However, it is hard to analyze system dynamics just looking at the changes in these variables, as causal relationships between variables are not readily apparent. By removing all the details about the actual state and the actual temporal information, we can view the dynamics as a graph with nodes describing states and links describing transitions. For instance software applications can have a large number of states. Problems occur when software applications reach uncommon or unanticipated states. Being able to visualize the entire state space, and quickly comprehend the paths leading to any particular state, allows more targeted analysis. Common states can be thoroughly tested, uncommon states can be identified and artificially induced. State space diagrams allow for numerous insights into system behaviour, in particular some states of the system can be shown to be unreachable, while others are unavoidable. Its applicability lies in any situation in which you have a model or system which changes state over time and you want to examine the abstract dynamical qualities of these changes. For example, social network theory, gene regulatory networks, urban and agricultural water usage, and concept maps in cognition and language modeling.

In such a scenario, every state of the system would be represented by a unique point in the state-space and the dynamics of the system would be mapped by trajectories through the state-space. These trajectories when converge at a point, are said to converge in on a basin of attraction, or simply an attractor, and it is at this point that any system reaches stability.

ecplex_basin

But, would this attractor phenomenon work for neural networks, where there are myriad nodes, each with their own corresponding state-spaces? The answer is in the affirmative, and thats because in stable systems, only a few attractor points are present, thus pulling in the system to stability. On the contrary, if the system is not stable, utter chaos would reign supreme, and this is where autopoiesis as a phenomenon comes to rescue by balancing perfectly between chaos and ordered states. This balancing act is extremely crucial and sensitive, for on the one hand, a chaotic system is too disordered to be beneficial, and on the other, a system that is highly stable suffers a handicap in dedicating a lot of resources towards reaching and maintaining attractor point/s. Not only that, even a transition from one stable state to another would rope in sluggish responses in adaptability to environment, and that too at a heavy cost of perturbations. But, self-organizing criticality would take care of such high costs by optimally utilizing available resources. And as the nodes are in possession of unequal weights to begin with, the fight for superiority takes precedence, that gets reflected in the state-space as well. If inputs are marked by variations, optimization through autopoietic criticality takes over, else, the system settles down to a strong attractor/s.

The nodes that interact at local zones are responsible for the macro-level effects, and according to Kauffman, this is possible in simple networks, by the switching values of “on” or “off” at input. In such cases, order is ensured with the formation of cores of stability that thereafter permeate the network and further see to it that the system reaches stability by drawing in other nodes into stability. In complex networks, nonlinearity is incorporated to adjust signals approaching critical point. The adjustment sees to it that if the signals are getting higher values, the system as such would slide into stability, or otherwise into chaos. Therefore adjustment as a mechanism is an important factor in complex systems to self-organize by hovering around criticality, and this is what is referred to as “on the edge of chaos” by Lewin.

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