"A function is also sometimes referred to as a map or a mapping. This terminology is common in mathematics, but less so in physics or other scientific fields. The idea of a mapping is useful if one wants to think of a function as acting on an entire set of input values." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"Chaos does provide a framework or a mindsetor point of view, but it is not as directly explanatory as germ theory or plate tectonics. Chaos is a behavior - a phenomenon - not a causal mechanism. [...] The situation with fractals is similar. The study of fractals draws one’s eye toward patterns and structures that repeat across different length or time scales. There is also a set of analytical tools - mainly calculating various fractal dimensions - that can be used to quantify structural properties of fractals. Fractal dimensions and related quantities have become standard tools used across the sciences. As with chaos, there is not a fractal theory. However, the study of fractals has helped to explain why certain types of shapes and patterns occur so frequently." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"Chaos is a phenomenon encountered in science and mathematics wherein a deterministic (rule-based) system behaves unpredictably. That is, a system which is governed by fixed, precise rules, nevertheless behaves in a way which is, for all practical purposes, unpredictable in the long run. The mathematical use of the word 'chaos' does not align well with its more common usage to indicate lawlessness or the complete absence of order. On the contrary, mathematically chaotic systems are, in a sense, perfectly ordered, despite their apparent randomness. This seems like nonsense, but it is not." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"Chaos is just one phenomenon out of many that are encountered in the study of dynamical systems. In addition to behaving chaotically, systems may show fixed equilibria, simple periodic cycles, and more complicated behaviors that defy easy categorization. The study of dynamical systems holds many surprises and shows that the relationships between order and disorder, simplicity and complexity, can be subtle, and counterintuitive." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"Fractals are different from chaos. Fractals are self-similar geometric objects, while chaos is a type of deterministic yet unpredictable dynamical behavior. Nevertheless, the two ideas or areas of study have several interesting and important links. Fractal objects at first blush seem intricate and complex. However, they are often the product of very simple dynamical systems. So the two areas of study - chaos and fractals - are naturally paired, even though they are distinct concepts." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"Functions are the most basic way of mathematically representing a relationship. [...] In mathematics, it is useful to think of a function as an action; a function takes a number as input, does something to it, and outputs a new number. [...] We are now in a position to refine our definition of a function. A function is a rule that assigns an output value f(x) to every input x. This is consistent with the everyday use of the word function: the output f(x) is a function of the input x. The output depends on the input [...]" (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"It is difficult to precisely define what is meant by a scientific theory, but usually a theory is a concise and consistent body of knowledge that allows one to understand a broad class of natural phenomena. [...] A scientific theory can provide a broad explanatory framework without being associated with the equations and calculational methods that are typical of physics." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"The study of chaos shows that simple systems can exhibit complex and unpredictable behavior. This realization both suggests limits on our ability to predict certain phenomena and that complex behavior may have a simple explanation. Fractals give scientists a simple and concise way to qualitatively and quantitatively understand self-similar objects or phenomena. More generally, the study of chaos and fractals hold many fun surprises; it challenges one’s intuition about simplicity and complexity, order and disorder." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"The study of fractals gives us a quantitative language to describe the myriad of self-similar shapes found in the natural world, including: mountain ranges; river basins; clouds and lightning; trees, ferns, and other plants; and vascular systems in plants and animals. Fractals need not be natural objects; they can be human-made and can also unfold in time in addition to space." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"The term 'chaos' has somewhat of a dual life. It refers to a particular type of dynamical behavior, but it is also often used as a general shorthand for the study of dynamical systems." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"Thus, much of physics can be seen as a iterative process: an object or a bunch of objects have some initial condition or seed. [...] Iterated functions are an example of what mathematicians call dynamical systems. A dynamical system is just a generic name for some variable or set of variables that change over time. There are many different types of dynamical systems - the iterated functions introduced above are just one type among many. Dynamical systems is now generally recognized as a branch of applied mathematics that studies properties of how systems change over time." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"Understanding chaos requires much less advanced mathematics than other current areas of physics research such as general relativity or particle physics. Observing chaos and fractals requires no specialized equipment; chaos is seen in scores of everyday phenomena - a boiling pot of water, a dripping faucet, shifting weather patterns. And fractals are almost ubiquitous in the natural world. Thus, it is possible to teach the central ideas and insights of chaos in a rigorous, genuine, and relevant way to students with relatively little mathematics background." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"A dynamical system is any mathematical system that changes
in time according to a well specified rule."
"Typically, in a mathematical model or the real world, one
only expects to observe stable fixed points. An unstable fixed point is
susceptible to a small perturbation; a tiny external influence will move the system
away from the unstable fixed point." (David P Feldman, "Chaos and Dynamical
Systems", 2019)
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