On Linearity II

"Why are nonlinear systems so much harder to analyze than linear ones? The essential difference is that linear systems can be broken down into parts. Then each part can be solved separately and finally recombined to get the answer. This idea allows a fantastic simplification of complex problems, and underlies such methods as normal modes, Laplace transforms, superposition arguments, and Fourier analysis. In this sense, a linear system is precisely equal to the sum of its parts." (Steven H Strogatz, "Non-Linear Dynamics and Chaos, 1994)

"When we examine the modeling literature, its most striking aspect is the predominance of 'flat' linear models. Why is this the case? After all, from a singularity theory viewpoint these linear objects are mathematical rarities. On mathematical grounds we should certainly not expect to see them put forth as credible representations of reality. Yet they are. And the reason is simple: linearity is a neutral assumption that leads to mathematically tractable models. So unless there is good reason to do otherwise, why not use a linear model?"  (John L Casti, "Five Golden Rules", 1995)

"In a linear system a tiny push produces a small effect, so that cause and effect are always proportional to each other. If one plotted on a graph the cause against the effect, the result would be a straight line. In nonlinear systems, however, a small push may produce a small effect, a slightly larger push produces a proportionately larger effect, but increase that push by a hair’s breadth and suddenly the system does something radically different." (F David Peat, "From Certainty to Uncertainty", 2002)

"Linearity means that the rule that determines what a piece of a system is going to do next is not influenced by what it is doing now. More precisely, this is intended in a differential or incremental sense: For a linear spring, the increase of its tension is proportional to the increment whereby it is stretched, with the ratio of these increments exactly independent of how much it has already been stretched. Such a spring can be stretched arbitrarily far, and in particular will never snap or break. Accordingly, no real spring is linear." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004) 

"Most long-range forecasts of what is technically feasible in future time periods dramatically underestimate the power of future developments because they are based on what I call the 'intuitive linear' view of history rather than the 'historical exponential' view." (Ray Kurzweil, "The Singularity is Near", 2005)

"Linear systems do not benefit from noise because the output of a linear system is just a simple scaled version of the input [...] Put noise in a linear system and you get out noise. Sometimes you get out a lot more noise than you put in. This can produce explosive effects in feedback systems that take their own outputs as inputs." (Bart Kosko, "Noise", 2006)

"On a linear system like a scale, the whole is equal to the sum of the parts. That’s the first key property of linearity. The second is that causes are proportional to effects. […] These two properties - the proportionality between cause and effect, and the equality of the whole to the sum of the parts - are the essence of what it means to be linear. […] The great advantage of linearity is that it allows for reductionist thinking. To solve a linear problem, we can break it down to its simplest parts, solve each part separately, and put the parts back together to get the answer." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"[...] perhaps one of the most important features of complex systems, which is a key differentiator when comparing with chaotic systems, is the concept of emergence. Emergence 'breaks' the notion of determinism and linearity because it means that the outcome of these interactions is naturally unpredictable. In large systems, macro features often emerge in ways that cannot be traced back to any particular event or agent. Therefore, complexity theory is based on interaction, emergence and iterations." (Luis Tomé & Şuay Nilhan Açıkalın, "Complexity Theory as a New Lens in IR: System and Change" [in "Chaos, Complexity and Leadership 2017", Şefika Şule Erçetin & Nihan Potas], 2019)

"With a linear growth of errors, improving the measurements could always keep pace with the desire for longer prediction. But when errors grow exponentially fast, a system is said to have sensitive dependence on its initial conditions. Then long-term prediction becomes impossible. This is the philosophically disturbing message of chaos." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)