"It is sometimes possible to infer a great deal about a dynamical system simply by finding its equilibrium states and determining which of these are stable to small disturbances and which are unstable. This can also help explain sudden or 'catastrophic' jumps from one state to another as some parameter is gradually varied." (David Acheson, "From Calculus to Chaos: An Introduction to Dynamics", 1997)
"Real dynamical problems typically involve nonlinear
differential equations of second order, but these often simplify greatly if we
investigate small oscillations about a position of equilibrium. Coupled
oscillators are particularly interesting, an early example being the double
pendulum, first studied by Euler and Daniel Bernoulli in the 1730s."
"Systems which exhibit chaotic oscillations typically do so
for some ranges of the relevant parameters but not for others, so one matter of
obvious interest is how the chaos appears (or disappears) as one of the
parameters is gradually varied."
"While calculus is the mathematical key to an understanding of Nature, its roots lie really in problems of geometry." (David Acheson, "From Calculus to Chaos: An Introduction to Dynamics", 1997)
"Differential equations provide, then, some of the deepest links between mathematics and the physical world." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)
"[…] it is all too easy in mathematics to jump to the wrong conclusion. And it is particularly dangerous to jump to some general conclusion on the basis of a few special cases." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)
"So, when trying to solve a problem in mathematics we have to watch out for subtle mistakes, otherwise, we can easily get the wrong solution." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)
"[…] the branch of mathematics which is most concerned with change is calculus. The key idea of calculus is in fact not so much change itself, but rather the rate at which change occurs." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)
"This, then, is the essence of chaos: irregular, erratic motion which is extremely sensitive to the initial conditions. […] A hallmark of chaos: two almost imperceptibly different starting conditions lead to two completely different outcomes, within a relatively short space of time." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)
"This general kind of behaviour, where a gradual change in
some parameter can lead to a sudden and unexpected large change in the system
as a whole, is known as a catastrophe."
"Today, the whole subject of geometry extends way beyond the world of right-angled triangles, circles and so on. There are even branches of the subject in which the ideas of length, angle and area don’t really feature at all. One of these is topology – a sort of rubber-sheet geometry – where a recurring question is whether some geometric object can be deformed ‘smoothly’ into another one." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)
"While any one of us is fully entitled – of course – to a quite different opinion, this amazing connection between e, i and π is viewed by many mathematicians as, quite simply, the most stunning result in the whole subject … so far." (David Acheson, "1089 and All That: A Journey into Mathematics", 2002)
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