“A model is a useful (and often indispensable) framework on which to organize our knowledge about a phenomenon. […] It must not be overlooked that the quantitative consequences of any model can be no more reliable than the a priori agreement between the assumptions of the model and the known facts about the real phenomenon. When the model is known to diverge significantly from the facts, it is self-deceiving to claim quantitative usefulness for it by appeal to agreement between a prediction of the model and observation.” (John R Philip, 1966)
"The flapping of a single butterfly’s wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done." (Ian Stewart, "Does God Play Dice?", 1989)
"Understandably, invariant sets (and their complements) play a crucial role in dynamic systems in general because they tell the most important fact about any initial condition, namely, its eventual fate: will the iterates be bounded, or will they be unstable and diverge? Or will the orbit be periodic or aperiodic?" (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)
"Systems, acting dynamically, produce (and incidentally, reproduce) their own boundaries, as structures which are complementary (necessarily so) to their motion and dynamics. They are liable, for all that, to instabilities chaos, as commonly interpreted of chaotic form, where nowadays, is remote from the random. Chaos is a peculiar situation in which the trajectories of a system, taken in the traditional sense, fail to converge as they approach their limit cycles or 'attractors' or 'equilibria'. Instead, they diverge, due to an increase, of indefinite magnitude, in amplification or gain."
"There are a variety of swarm topologies, but the only organization that holds a genuine plurality of shapes is the grand mesh. In fact, a plurality of truly divergent components can only remain coherent in a network. No other arrangement-chain, pyramid, tree, circle, hub-can contain true diversity working as a whole. This is why the network is nearly synonymous with democracy or the market." (Kevin Kelly, "Out of Control: The New Biology of Machines, Social Systems and the Economic World", 1995)
"In a complex system, it is not uncommon for subsystems to have goals that compete directly with or diverge from the goals of the overall system. […] Feedback gathered from small, local subsystems for use by larger subsystems may be either inaccurately conveyed or inaccurately interpreted. Yet it is this very flexibility and looseness that allow large, complex systems to endure, although it can be hard to predict what these organizations are likely to do next." (Virginia Anderson & Lauren Johnson, "Systems Thinking Basics: From Concepts to Causal Loops", 1997)
"A depressing corollary of the butterfly effect (or so it was widely believed) was that two chaotic systems could never synchronize with each other. Even if you took great pains to start them the same way, there would always be some infinitesimal difference in their initial states. Normally that small discrepancy would remain small for a long time, but in a chaotic system, the error cascades and feeds on itself so swiftly that the systems diverge almost immediately, destroying the synchronization. Unfortunately, it seemed, two of the most vibrant branches of nonlinear science - chaos and sync - could never be married. They were fundamentally incompatible." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"Chaotic system is a deterministic dynamical system exhibiting irregular, seemingly random behavior. Two trajectories of a chaotic system starting close to each other will diverge after some time (such an unstable behavior is often called 'sensitive dependence on initial conditions'). Mathematically, chaotic systems are characterized by local instability and global boundedness of the trajectories. Since local instability of a linear system implies unboundedness (infinite growth) of its solutions, chaotic system should be necessarily nonlinear, i.e., should be described by a nonlinear mathematical model." (Alexander L Fradkov, "Cybernetical Physics: From Control of Chaos to Quantum Control", 2007)
"Complexity theory shows that great changes can emerge from small actions. Change involves a belief in the possible, even the 'impossible'. Moreover, social innovators don’t follow a linear pathway of change; there are ups and downs, roller-coaster rides along cascades of dynamic interactions, unexpected and unanticipated divergences, tipping points and critical mass momentum shifts. Indeed, things often get worse before they get better as systems change creates resistance to and pushback against the new. Traditional evaluation approaches are not well suited for such turbulence. Traditional evaluation aims to control and predict, to bring order to chaos. Developmental evaluation accepts such turbulence as the way the world of social innovation unfolds in the face of complexity. Developmental evaluation adapts to the realities of complex nonlinear dynamics rather than trying to impose order and certainty on a disorderly and uncertain world." (Michael Q Patton, "Developmental Evaluation", 2010)
"The key characteristic of 'chaotic solutions' is their sensitivity to initial conditions: two sets of initial conditions close together can generate very different solution trajectories, which after a long time has elapsed will bear very little relation to each other. Twins growing up in the same household will have a similar life for the childhood years but their lives may diverge completely in the fullness of time. Another image used in conjunction with chaos is the so-called 'butterfly effect' – the metaphor that the difference between a butterfly flapping its wings in the southern hemisphere (or not) is the difference between fine weather and hurricanes in Europe." (Tony Crilly, "Fractals Meet Chaos" [in "Mathematics of Complexity and Dynamical Systems"], 2012)
"The most basic tenet of chaos theory is that a small change in initial conditions - a butterfly flapping its wings in Brazil - can produce a large and unexpected divergence in outcomes - a tornado in Texas. This does not mean that the behavior of the system is random, as the term 'chaos' might seem to imply. Nor is chaos theory some modern recitation of Murphy’s Law ('whatever can go wrong will go wrong'). It just means that certain types of systems are very hard to predict." (Nate Silver, "The Signal and the Noise: Why So Many Predictions Fail-but Some Don't", 2012)
"Chaos is a long-term behavior of a nonlinear dynamical system that never falls in any static or periodic trajectories. [It] looks like a random fluctuation, but still occurs in completely deterministic, simple dynamical systems. [It] exhibits sensitivity to initial conditions. [It] occurs when the period of the trajectory of the system’s state diverges to infinity. [It] occurs when no periodic trajectories are stable." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)
"The failures of economics in the practical world are largely due to seeing the economy in equilibrium. […] Equilibrium thinking cannot 'see' such exploitation in advance for a subtle reason: by definition, equilibrium is a condition where no agent has any incentive to diverge from its present behavior, therefore exploitive behavior cannot happen. And it cannot see extreme market behavior easily either: divergences are quickly corrected by countervailing forces. By its base assumptions, equilibrium economics is not primed to look for exploitation of parts of the economy or for system breakdowns." (W Brian Arthur, "Complexity and the Economy", 2015)
"Trajectories of a deterministic dynamical system will never branch off in its phase space (though they could merge), because if they did, that would mean that multiple future states were possible, which would violate the deterministic nature of the system. No branching means that, once you specify an initial state of the system, the trajectory that follows is uniquely determined too. You can visually inspect where the trajectories are going in the phase space visualization. They may diverge to infinity, converge to a certain point, or remain dynamically changing yet stay in a confined region in the phase space from which no outgoing trajectories are running out. Such a converging point or a region is called an attractor." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)
"A system in which a few things interacting produce tremendously divergent behavior; deterministic chaos; it looks random but its not." (Christopher Langton)
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