"[The] system may evolve through a whole succession of transitions leading to a hierarchy of more and more complex and organized states. Such transitions can arise in nonlinear systems that are maintained far from equilibrium: that is, beyond a certain critical threshold the steady-state regime become unstable and the system evolves into a new configuration." (Ilya Prigogine, Gregoire Micolis & Agnes Babloyantz, "Thermodynamics of Evolution", Physics Today 25" (11), 1972)
"As the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics." (Lotfi A Zadeh, 1973)
"Fuzziness, then, is a concomitant of complexity. This implies that as the complexity of a task, or of a system for performing that task, exceeds a certain threshold, the system must necessarily become fuzzy in nature. Thus, with the rapid increase in the complexity of the information processing tasks which the computers are called upon to perform, we are reaching a point where computers will have to be designed for processing of information in fuzzy form. In fact, it is the capability to manipulate fuzzy concepts that distinguishes human intelligence from the machine intelligence of current generation computers. Without such capability we cannot build machines that can summarize written text, translate well from one natural language to another, or perform many other tasks that humans can do with ease because of their ability to manipulate fuzzy concepts." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic", 1989)
"In the realms of nature it is impossible to predict which way a bifurcation will cut. The outcome of a bifurcation is determined neither by the past history of a system nor by its environment, but only by the interplay of more or less random fluctuations in the chaos of critical destabilization. One or another of the fluctuations that rock such a system will suddenly 'nucleate'. The nucleating fluctuation will amplify with great rapidity and spread to the rest of the system. In a surprisingly short time, it dominates the system’s dynamics. The new order that is then born from the womb of chaos reflects the structural and functional characteristics of the nucleated fluctuation. [...] Bifurcations are more visible, more frequent, and more dramatic when the systems that exhibit them are close to their thresholds of stability - when they are all but choked out of existence." (Ervin László, "Vision 2020: Reordering Chaos for Global Survival", 1994)
"When a system is 'stressed' beyond certain threshold limits as, for example, when it is heated up, or its pressure is increased, it shifts from one set of attractors to another and then behaves differently. To use the language of the theory, the system 'settles into a new dynamic regime'. It is at the point of transition that a bifurcation takes place. The system no longer follows the trajectory of its initial attractors, but responds to new attractors that make the system appear to be behaving randomly. It is not behaving randomly, however, and this is the big shift in our understanding caused by dynamical systems theory. It is merely responding to a new set of attractors that give it a more complex trajectory. The term bifurcation, in its most significant sense, refers to the transition of a system from the dynamic regime of one set of attractors, generally more stable and simpler ones, to the dynamic regime of a set of more complex and 'chaotic' attractors." (Ervin László, "Vision 2020: Reordering Chaos for Global Survival", 1994)
"For any given population of susceptibles, there is some critical combination of contact frequency, infectivity, and disease duration just great enough for the positive loop to dominate the negative loops. That threshold is known as the tipping point. Below the tipping point, the system is stable: if the disease is introduced into the community, there may be a few new cases, but on average, people will recover faster than new cases are generated. Negative feedback dominates and the population is resistant to an epidemic. Past the tipping point, the positive loop dominates .The system is unstable and once a disease arrives, it can spread like wildfire that is, by positive feedback-limited only by the depletion of the susceptible population." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)
"In the case of a complex system, nonlinear behavior can happen as disturbances or changes in the system, each one relatively small by itself, accumulate. Outwardly, everything seems to be normal: the system doesn’t generate any surprises. At some point, though, the behavior of the whole system suddenly shifts to a radically new mode. This kind of behavior is often called a threshold effect, because the shift occurs when a critical threshold - usually unseen and often unexpected - is crossed." (Thomas Homer-Dixon, "The Upside of Down: Catastrophe, Creativity, and the Renewal of Civilization", 2006)
"Creative elements add random elements to network behavior, inducing an increase in noise. This is highly beneficial to a certain extent as we saw in the previous box, but becomes intolerable if it exceeds a certain threshold. This threshold is high if the hosting network lives an individual life and often meets unexpected situations. However, the same threshold becomes low if the hosting network is part of a higher level organization which provides a stable environment." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"The simplest basic architecture of an artificial neural network is composed of three layers of neurons - input, output, and intermediary" (historically called perceptron). When the input layer is stimulated, each node responds in a particular way by sending information to the intermediary level nodes, which in turn distribute it to the output layer nodes and thereby generate a response. The key to artificial neural networks is in the ways that the nodes are connected and how each node reacts to the stimuli coming from the nodes it is connected to. Just as with the architecture of the brain, the nodes allow information to pass only if a specific stimulus threshold is passed. This threshold is governed by a mathematical equation that can take different forms. The response depends on the sum of the stimuli coming from the input node connections and is 'all or nothing'." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)
"Even more important is the way complex systems seem to strike a balance between the need for order and the imperative for change. Complex systems tend to locate themselves at a place we call 'the edge of chaos'. We imagine the edge of chaos as a place where there is enough innovation to keep a living system vibrant, and enough stability to keep it from collapsing into anarchy. It is a zone of conflict and upheaval, where the old and new are constantly at war. Finding the balance point must be a delicate matter - if a living system drifts too close, it risks falling over into incoherence and dissolution; but if the system moves too far away from the edge, it becomes rigid, frozen, totalitarian. Both conditions lead to extinction. […] Only at the edge of chaos can complex systems flourish. This threshold line, that edge between anarchy and frozen rigidity, is not a like a fence line, it is a fractal line; it possesses nonlinearity." (Stephen H Buhner, "Plant Intelligence and the Imaginal Realm: Beyond the Doors of Perception into the Dreaming of Earth", 2014)
"Flaws can be found in any research design if you look hard enough. […] In our experience, it is good scientific practice to refine one's research hypotheses in light of the data. Working scientists are also keenly aware of the risks of data dredging, and they use confidence intervals and p-values as a tool to avoid getting fooled by noise. Unfortunately, a by-product of all this struggle and care is that when a statistically significant pattern does show up, it is natural to get excited and believe it. The very fact that scientists generally don't cheat, generally don't go fishing for statistical significance, makes them vulnerable to drawing strong conclusions when they encounter a pattern that is robust enough to cross the p < 0.05 threshold." (Andrew Gelman & Eric Loken, "The Statistical Crisis in Science", American Scientist Vol. 102(6), 2014)
"Even more important is the way complex systems seem to strike a balance between the need for order and the imperative for change. Complex systems tend to locate themselves at a place we call 'the edge of chaos'. We imagine the edge of chaos as a place where there is enough innovation to keep a living system vibrant, and enough stability to keep it from collapsing into anarchy. It is a zone of conflict and upheaval, where the old and new are constantly at war. Finding the balance point must be a delicate matter - if a living system drifts too close, it risks falling over into incoherence and dissolution; but if the system moves too far away from the edge, it becomes rigid, frozen, totalitarian. Both conditions lead to extinction. […] Only at the edge of chaos can complex systems flourish. This threshold line, that edge between anarchy and frozen rigidity, is not a like a fence line, it is a fractal line; it possesses nonlinearity." (Stephen H Buhner, Plant Intelligence and the Imaginal Realm: Beyond the Doors of Perception into the Dreaming of Earth", 2014)
"Only at the edge of chaos can complex systems flourish. This threshold line, that edge between anarchy and frozen rigidity, is not a like a fence line, it is a fractal line; it possesses nonlinearity." (Stephen H Buhner, "Plant Intelligence and the Imaginal Realm: Beyond the Doors of Perception into the Dreaming of Earth", 2014)
"Bifurcation is a qualitative, topological change of a system’s phase space that occurs when some parameters are slightly varied across their critical thresholds. Bifurcations play important roles in many real-world systems as a switching mechanism. […] There are two categories of bifurcations. One is called a local bifurcation, which can be characterized by a change in the stability of equilibrium points. It is called local because it can be detected and analyzed only by using localized information around the equilibrium point. The other category is called a global bifurcation, which occurs when non-local features of the phase space, such as limit cycles (to be discussed later), collide with equilibrium points in a phase space. This type of bifurcation can’t be characterized just by using localized information around the equilibrium point." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)
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