"Up until now most economists have concerned themselves with linear systems, not because of any belief that the facts were so simple, but rather because of the mathematical difficulties involved in nonlinear systems [... Linear systems are] mathematically simple, and exact solutions are known. But a high price is paid for this simplicity in terms of special assumptions which must be made." (Paul A Samuelson, "Foundations of Economic Analysis", 1966)
"Linear relationships are easy to think about: the more the merrier. Linear equations are solvable, which makes them suitable for textbooks. Linear systems have an important modular virtue: you can take them apart and put them together again - the pieces add up. Nonlinear systems generally cannot be solved and cannot be added together. [...] Nonlinearity means that the act of playing the game has a way of changing the rules. [...] That twisted changeability makes nonlinearity hard to calculate, but it also creates rich kinds of behavior that never occur in linear systems." (James Gleick, "Chaos: Making a New Science", 1987)
"Never in the annals of science and engineering has there been a phenomenon so ubiquitous‚ a paradigm so universal‚ or a discipline so multidisciplinary as that of chaos. Yet chaos represents only the tip of an awesome iceberg‚ for beneath it lies a much finer structure of immense complexity‚ a geometric labyrinth of endless convolutions‚ and a surreal landscape of enchanting beauty. The bedrock which anchors these local and global bifurcation terrains is the omnipresent nonlinearity that was once wantonly linearized by the engineers and applied scientists of yore‚ thereby forfeiting their only chance to grapple with reality." (Leon O Chua, "Editorial", International Journal of Bifurcation and Chaos, Vol. l (1), 1991)
"It remains an unhappy fact that there is no best method for finding the solution to general nonlinear optimization problems. About the best general procedure yet devised is one that relies upon imbedding the original problem within a family of problems, and then developing relations linking one member of the family to another. If this can be done adroitly so that one family member is easily solvable, then these relations can be used to step forward from the solution of the easy problem to that of the original problem. This is the key idea underlying dynamic programming, the most flexible and powerful of all optimization methods." (John L Casti, "Five Golden Rules", 1995)
"When it comes to modeling processes that are manifestly governed by nonlinear relationships among the system components, we can appeal to the same general idea. Calculus tells us that we should expect most systems to be 'locally' flat; that is, locally linear. So a conservative modeler would try to extend the word 'local' to hold for the region of interest and would take this extension seriously until it was shown to be no longer valid." (John L Casti, "Five Golden Rules", 1995)
"A system at a bifurcation point, when pushed slightly, may begin to oscillate. Or the system may flutter around for a time and then revert to its normal, stable behavior. Or, alternatively it may move into chaos. Knowing a system within one range of circumstances may offer no clue as to how it will react in others. Nonlinear systems always hold surprises."
"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."
"Complex systems are full of interdependencies - hard to
detect - and nonlinear responses." (Nassim N Taleb, "Antifragile: Things That
Gain from Disorder", 2012)
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