Mathematics is just a method, and one of its main characteristics is abstractness.Abstraction's perhaps the single most important word for appreciatingmathematical work, have it some natural phenomena as an amorphous andrapidly abandoned spawn or not even that. Grammatically, the root formis the adjectival, from the Latin abstractus = "drawn away". Some major (english) dictionary definitions of interest(1):– "withdrawn/separated from matter/material embodiment/practice/particular examples;" – "ideal, distilled to its essence." Then,for our purposes, abstract's basic adopted definition will be twofold:a) (adjective) removed from or transcending concrete particularity,sensuous experience; (b) (verb) abstract something = reducing it (e.g.,a natural/fabricated phenomenum – industrial, biological, sociological,metheorological, etc., in nature, at every conceivable level) to itsabsolute skeletal essence(2), frequently as the only practical way toproduce some minimum understanding of an extremelly complex phenomenumpretty sure (in)directly affected by a huge number of influences atvarious different degrees. Mathematicians and nowadays also asignificant proportion of biologists, sociologists, economists et al.,as well as, you know, philosophers, spend a lot of time thinkingabstractly, or about abstractions, or both. Abstraction proceeds inlevels. For instance, let's say man meaning some particular man is Level One, Man meaning the species is Level Two, something like humanity/humannessis Level Three, and so forth. Quoting Bertrand Russel: "That fact is, that in Algebra the mind isfirst taught to consider general truths no asserted to hold only ofthis or that particular thing, but of any one of a whole group ofthings. It's in the power of understanding/discovering such truths thatthe mastery of the intellect of the whole world of things actual &possible resides; and ability to deal with the general as such is oneof the gifts a mathematical education should bestow." In a more purposeful instance, consider a simple pendulum (Figure 1), an idealized (physical) system composed of an object of m units of mass and no particularly relevant shape + a perfectly rigid l-unitlong stem with almost 0 units of mass (i.e., "no mass" for praticalpurposes) who freely moves along a perfect, single vertical plane. Notethat the general variables, or simply noums, m for mass and lfor height, are only mathematical entities entertained by the mind,general concepts divorced from particular instances. As well as motion.What exactly do we mean when we talk about motion? In the particularlevel of abstraction embedded in our example motion corresponds toangular displacement, k, along a single (vertical) plane, of a general object of mass m, connected to the end of a fixed, perfectly rigid stem of height l.If we abandon the object from a particular (angular) position, thependular motion starts. Consider that we have ideally assumed that thisis a free motion, meaning that the (supposed) air friction wasabstracted out (except the laws of gravitation). As time passes (andlet's not begin, given our current more modest purposes, a discussionabout the most important abstraction of all abstractions) on, theobject of mass m (henceforth simply m) moves along theplane theoretically assuming all possible angular displacements,provided that we consider infinitesimally small time steps (i.e.,almost 0), until a cessation determined only by its mass and the stemheight. Let's call the abstract "mode" where we can monitor m's position (i.e., observe and perfectly measure and collect k) within almost 0 time steps a continuous mode, or simply continuous time.Reality dictates that we can only register those angular displacementswithin significant time steps determined by the technology we use to dothe measurements, who actually samples the first (idealized) set ofcollected k's. Let's call this "mode" discrete time(3)(4).
Given all such idealized conditions we can perhaps mathematicallyrepresent this specific, simple pendular movement by an equation such as
A: m x l2 x (dk2/d2t) + m x g x l x Sin(k) = 0where k is the pendulum angular displacement, dk2/d2t is the second (temporal) derivative of k, and g is the well-known local acceleration due to gravity or standard gravity (value = 9.80665 m/s2).A derivative is defined by the ratio of variation of one or morevariables in relation to the variation of another one or morevariables. In its most simple form, we have a derivative of a singlevariable in relation to time, or, in our particular example, thevariation of a single variable, k, in relation to only onevariable, time, considering the ideal case where time steps areinfinitesimally small, an operation we usually symbolically representby dk/dt, meaning the first temporal derivative of k. (Abstractly) Repeating this same (mathematical) process n times we have an n-th order temporal derivative of k. Now note the second term in the left-hand side of equation A: Sin(k) is the sine of k, the well-known trigonometrical function – i.e., for each value of k Sin(k)returns a particular value exactly equal to the sine of that angle.Take note that usually we can also derivate a function in relation totime or in relation to one (or more) of its constituent variables. So A is a second-order Ordinary Differential Equation(henceforth O.D.E.), "ordinary" in the sense that it contains functionsof only one independent variable, and one or more of its derivativeswith respect to that variable. O.D.E.s are one of the most common andsimple building blocks for the mathematical representation of actualphysical/biological/sociological/economical/… phenomema. In fact,there are an enormous number of these abstract representationsavailable to the analyst. Let's call them simply mathematical models.The most high level distinction between them is the following: Modelsas in A are constructed based on first-principles knowledge of the system's underlying physical laws, while there are models that are pure unspecific black-boxes with several internal characteristics to be somewhat adjusted based on recorded information of system's behavior. Despitethe very idealized conditions, there's no analytical way to solve A interms of elementary functions(5). However, we can hopefully identifythe main characteristics of those "unreachable" solutions,qualitativelly scanning the possible movements inherent to this quiteidealized representation of the actual physical system. The simple pendulum as in A is an autonomous system,since there's no explicit dependence on time. In a non-autonomoussystem, behavior explicitly depends on time. Think of a pendulum that'speriodically forced by an external stimuli, in which case we canidentify a system output – the angular displacement – and a systeminput – the external force. I.e., there's a direct causal relationbetween the external stimuli & the angular displacement. A systemis called causal if the output at any time depends only on values ofthe input at the present time and/or in the past. Embedded in thisdefinition's another important one, that of a dynamical system– i.e., if a system's behavior – e.g., represented by its identifiedoutput – also somewhat depends on the past values of this output and/orpast values of the input, that system is not a static, memoryless onefor which current inputs are all that's needed to determine the currentoutputs. Roughly speaking, the concept of dynamics in a systemcorresponds to the presence of an internal "mechanism" thatretains/stores information about input and/or output values at timesother than the current time. In some physical systems, memory'sdirectly associated with the storage of some form of energy – e.g.,think of a simple electrical circuit with a capacitor who stores energyby accumulating electrical charge. An automobile has memory stored inits kinetic energy. In computers, memory's typically directlyassociated with storage registers that retain values between clockpulses. At this point is quite natural to recognize almost allconceivable system or phenomenum as dynamic by nature. However, there'ssome useful cases where this memory effect can be small enough to makea simple static (mathematical) representation of the system all that'sneeded for the majority of meaningful purposes. There's also an inherently nonlinear behavior embedded in the second term of the left-hand side of A, as kwill depend of its own sine. A linear system possesses the property ofsuperposition: If an input consists of the weighted sum of severalsignals (i.e., recorded values of some variable evolving over time),then the output is the weighted sum of the responses of the system toeach of those signals. Nonlinear terms in the equations of a system caninvolve algebraic or more complicated functions and variables, andthese terms may have a physical couterpart, such as forces of inertiathat damp oscillations of a pendulum (e.g., air friction), viscosity ofa fluid, or the limits of growth of a biological population, to name afew. For the sake of simplicity, we can or must dismember thesecond-order O.D.E. in a set of two coupled first-order O.D.E.s byintroducing a new variable v = dk/dt. Actually,the first derivative (over time) of the variable "position" is(abstractly) another well-know variable called "velocity". So now wehave an (still) autonomous, 2-variable, 3-dimensions, 2-statemathematical model(6). Usually, the state of a system is a pairof variables (or a triple, whatever) who somewhat represents how thesystem behavior evolves over time. As a bidimensional representation,we can simply put k (position) and dk/dt (velocity) on the axes of a diagram on the plane and qualitatively sketch the phase paths, flow lines or orbits which represent A's possible solutions. Each pair (k,dk/dt) in the plane corresponds to a particular solution of A. Let's call the set of all these curves the system's phase portrait, see Figure 2. Note that point that the central point in the bottom part of the figure corresponds to A's trivial solution: k = 0 (then dk/dt = 0); physically, we put the object at k = 0 and the pendulum remains there indefinetelly, what we usually call an equilibrium point (or fixed point). There's another notable equilibrium point in Figure 2, b: k = pi, 3xpi, … and k = -pi, -3xpi, … and k= 0 (simply meaning that the pendulum is rotating clockwise oranti-clockwise), which physically corresponds to the object suspendedabove its stem, remaining there indefinetelly, obviously a prettycounter-intuitive condition. Let's make a more detailed look at the phase portrait in Figure 2. Theset of closed curves around fixed points represents all the supposedlyperiodical movements the simple pendulum can perform – i.e., as sometime passes, the system returns to the same state. Fixed points whereeach curve intersects axis k, corresponds to the oscillations amplitude. The wave-like curves at the top and at the bottom represent movements where k always increase or always decrease (conventionally, it's assumed that k increases clockwise) – i.e., movements where the pendulum rotates. The two curves who actually intersect axis kspatially define two separated regions with quite different(qualitative) behaviors: Inside these two curves, the movement isperiodical & bounded; outside, the movement is unbounded. As weabstracted out air friction and any other possible external influencebesides gravitation, the simple pendulum keeps its total energyindefinetelly – i.e., this hyper-idealized but yet somewhat useful(e.g., at least in an educational sense) mathematical representation ofan actual physical system (i.e., a pendulum) can be called aconvervative or Hamiltonian system. Let's go now to a moreinteresting yet idealized system, naturally ocurring or artificallybuilt, known as the damped harmonic oscillator, who we canmathematically idealize by the following second-order, autonomous,linear O.D.E. B: (dx2/d2t) + y x(dx/dt) + o2 x x = 0 This oscillator is a dissipative system, and note that, according toits phase portrait shown in Figure 3, there's a single point for whichall the trajectories converge provided that a sufficient long time haspassed – i.e., the movement is damped and finally comes to a halt. Thisvery important construct, only available for dissipative systems, isknown as the attractor of the system, which in this particularcase is only a point, because B is a linear equation, and coincideswith B's trivial solution, or the system's trivial equilibrium point -i.e. (x,dx/dt) = (0,0). That is, movementsapproach a fixed/equilibrium point, that attracts the closest orbits.However, as we'll see in the next parts of this "essay", attractorsmay have any dimension, provided that this dimension is smaller thanthe dimension of its system's equilibrium points (also known as thenumber of degrees of freedom).