"In fact, contrary to intuition, some of the most complicated dynamics arise from the simplest equations, while complicated equations often produce very simple and uninteresting dynamics. It is nearly impossible to look at a nonlinear equation and predict whether the solution will be chaotic or otherwise complicated. Small variations of a parameter can change a chaotic system into a periodic one, and vice versa." (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 2010)
"There are actually two sides to the success of mathematics in explaining the world around us (a success that Wigner dubbed ‘the unreasonable effectiveness of mathematics’), one more astonishing than the other. First, there is an aspect one might call ‘active’. When physicists wander through nature’s labyrinth, they light their way by mathematics - the tools they use and develop, the models they construct, and the explanations they conjure are all mathematical in nature. This, on the face of it, is a miracle in itself. […] But there is also a ‘passive’ side to the mysterious effectiveness of mathematics, and it is so surprising that the 'active' aspect pales by comparison. Concepts and relations explored by mathematicians only for pure reasons - with absolutely no application in mind - turn out decades (or sometimes centuries) later to be the unexpected solutions to problems grounded in physical reality!" (Mario Livio, "Is God a Mathematician?", 2011)
"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)
"Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values" (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations can occur in both continuous systems" (described by ODEs, DDEs, or PDEs) and discrete systems" (described by maps)." (Tianshou Zhou, "Bifurcation", 2013)
"Complex systems defy intuitive solutions. Even a third-order, linear differential equation is unsolvable by inspection. Yet, important situations in management, economics, medicine, and social behavior usually lose reality if simplified to less than fifth-order nonlinear dynamic systems. Attempts to deal with nonlinear dynamic systems using ordinary processes of description and debate lead to internal inconsistencies. Underlying assumptions may have been left unclear and contradictory, and mental models are often logically incomplete. Resulting behavior is likely to be contrary to that implied by the assumptions being made about' underlying system structure and governing policies." (Jay W Forrester, "Modeling for What Purpose?", The Systems Thinker Vol. 24 (2), 2013)
"Mathematics does not merely describe the problem in an abstract way, it allows us to find a previously unknown 'solution' from the abstract description. It is surprising that the unknown can be transformed into the well known when we succeed in describing the problem mathematically." (Waro Iwane, "Mathematics in Our Company: What Does It Describe?", [in "What Mathematics Can Do for You"] 2013)
"That’s where boundary conditions come in. A boundary condition 'ties down' a function or its derivative to a specified value at a specified location in space or time. By constraining the solution of a differential equation top satisfy the boundary condition(s), you may be able to determine the value of the function or its derivatives at other locations. We say “may” because boundary conditions that are not well-posed may provide insufficient or contradictory information." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)
"A transcendental number is defined as a number that isn’t the solution of any polynomial equation with integer constants times the x’s." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)
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