Why are Non-Linear Dynamical Systems Important ?

A thriller-like short essay, targeted to teenagers and YAs, which takes through a flight into a fundamental concept in modern life, science and technology

by Simon Hasur (alias 'the Nerd of Algorithms')


This short-essay is adressed practically to every type of Reader: it can, in fact, be understood at some level by anyone. The aim is utterly biased toward a young audience, indicatively from age 12 to 35 years ; but also older ages are fine. If something is not clear during the reading, or one realizes not having the necessary background-knowledge to understand that particular thing fully at a first reading, dont'worry! Just read the essay without problems... You can come back to re-read it at any time. Large part of this essay in fact, rather than telling a bunch of mere facts, tries instead to guide to get interested, and get to think about, some things which are extremely useful to have a more satisfactory life in our modern age. As far as the ability to achieve a deeper, or maybe full, understanding of what this little essay tries to convey, I trust in my Readers' intelligence, creativity, curiosity, and desire to live in a better today and an even better tomorrow. A Better Tomorrow to which, armed with some degree of knowledge of Non-Linear Dynamical Systems, They will likelily be also able to contribute actively.

In an epoch after the 2nd millenium in the standard historical time-classification, it can be crucial understanding Non-Linear Dynamical Systems, to live in harmony with them. As we are ubuquitously surrounded by lots of them. At close range, we are surrounded by living organisms including vegetation, people, and lots of other things which can be considered to a good or practically full extent, Non-Linear Dynamical Systems.

I bring to You this essay, dear Reader, to speak a bit about what these entities are, and why are they important. And why it's important in our modern epoch in which I am writing this essay, to have some degree of familiarity, or even some degree of friendship with them.

Two last introductory words:
1. I try to unfold this topic in the way of a thriller-like text, that be somewhere between an essay and a narration.
2. As the topic is also scientific, since scientific communication uses text, figures and formulas simulataneously, this one does so also. And it's in place to mention that this text has quite many well-tidied figures, whose creation has been possible and has been speeded up to optimal levels, thanks to a computer-program written by myself, in the C programming language: a program which, from simple commands about what needs to be represented, generates a text-file respecting the SVG syntax : a file which then opens and displays as a vector-graphics image.


What are Dynamical Systems

Good question, so let's keep it short... most doubts will dissolve as the discussion goes on. Pretty much anything that has a behaviour which is enough predictable to be possible for it to be simulated up to a good extent of fidelity. Something which needs to be contained, limited in volume, and remain always within a limited enclosing-volume: they are all isolated systems... one needs to be able to consider them so in order to classify them as Dynamical Systems. They are something one can interact with, and which responds to the stimuli through which one tries to interact: and the the interacion must be limited to a limited and well-defined interface (elctric contacts whose tension is measure, electric contacts whose voltage can be taken to a given voltage, or a certain electric of magnetic field in a specific spot, physical contact to apply a force to a point of a body, an idealized actuator, these kinds of things ) .
Moreover, Some Dynamical Systems produce an output by themselves, but the ability to interact with them and see that they change their output according to different input-stimuli, is important.
Electonic circuits, Mechanisms such as a swing, a pendulum, a vehicle, just any mechanism made of an utterly finite number of elements is so, thermic devices, a piece of magnetizable metal, a bipolar electronic component, a tripolar one, some air or a liquid closed in a rigid container, etc. Pretty much a lot of things that surround us in every-day life, are Dynamical Systems ; except living organisms (!). Chemical compounds contained in a very well controlled environment are limit-cases because they can change state, they can do lots of things which are difficult to measure, to control, etc: the solid-state ones which have some well-defined and very characteristic electric and magnetic properties, are instead fine.
One last thing is electromagnetic field: this is also at the limit, because it propagates... so it presents us with some difficulties in it's treatment, reason why this topic will be omitted even because I lack a sufficiently solid preparation on it.

Let this be enough.

A last word of introduction, is that some of these Dynamical Systems are much more simpler than others... and these are so simple that automatially get themselves categorized as ground-level in the study and simulation of the behaviour of Dynamical Systems in general.
These sorts of very dumb systems are called Linear Dynamical Systems... and there are also very few of them... and they look just pretty much all alike. Let's see some of them to get to the evident understanding that in modern times, we cannot limit our attention to these systems only. Useless to say that most of modern technology features at it's heart non-linear systems, and the linear ones are only there to accompany, to distribute, to deliver - so to speak - the core-work done by non-linear sub-systems.

All mass-spring-pulley mechanisms with transmission done by non-extensible wires with ideal full grip on the pulleys and having viscous dumping elements only (fig.1),

fig.1: a typical mass-spring-pulley mechansm, with transmission done by non-extensible wires with ideal full grip on the pulleys and having viscous dumping elements only.
figure missing.

are linear dynamical systems.

Masses can tied to be able to slip frictionlessly along either a straigth sustainment (fig.2),

fig.2: a) sraight sustainment to which a material-point
is constrained to slip along, frictionlessly
b) more points...
a) b)

either a perfecly circular one (fig.3) ;

fig.3: a) perfectly circular sustainment to which a material-point is constrained:
to slip along it, frictionlessly
b) more points and some springs:
combined to make up a chute little Linear Dynamical System

other curvilinear sustainments are not allowed, as they all lead to non-linear dynamical systems, in fact ( we'll see plenty of them later on ).
Here are for You another bunch of miscellaneous examples of linear dynamical systems, from the Mechanics' regime (fig.4).

fig.4: a) b)
a) b)
c) d)
c) a) figure d) missing.

All dynamical systems coming from the Mechanics' regime, have, before coming to the way they move, the way they change in time, a more basic feature: that their configuration at an instant can be defined by a finite, integer (obvious), number of numerical values ( these are continuous values, of course: consider them real, rational, or floating-point numbers according to the context ). Let's call these numerical values, parameters: the number indicating how many of these parameters are needed to univocously define a Dynamica System's configuration at a given time, is called the number of Degrees of Freedom( for short: DOFs ) which it has. Each of the previously mentioned parameters are, in the context of the definition, study and simulation of dynamical systems of the Mechanics' and a few other fields' regime, there are called generalized coordinates. There are as many generalized coordinates as many Degrees of Freedom the given system has.

So are Linear Dynamical Systems also all well-designed electronic circuits which feature only resistors, capacitors, and inductors (fig.5).

fig.5: a typical, linear electronic circuit's sheme.
figure missing.

The story about the number of Degrees of Freedom, is the seme.

Operational-Amplifiers are not allowed as they are designed to behave as linear, but at their heart there's heavily non-linear stuff as BJT transistors and diodes: plus, Operational-Amplifiers allow to construct a handful of non-linear circuits. This may sound as a common-place or a mere slogan, but as soon as one takes into account that with them we can construct dual-input tenstion-multipliers which give as output the product of the 2 tensions, it's pretty much done... multiplication of 2 parameters which define the state of a system, leads to a non-linear system... and this is only the simplest case of a non-linear system. This may appear impressing, it is indeed impressing and very important, but let's dont' start thinking negatively... we are here to familiarize, to make friendship with non-linear dynamical system, not to start hating them, avoid them, or downplay their role in the world in any way.

With this said, let's make a note here, before proceeding. In the mass-spring-pulley systems I was talking about a little before, there's anyway one special requirenment for them to fully classify as linear dynamical systems: they must have no springs which can display obliquously, that is non-parallelly, to a sustaining-rod to which some point is constrained (fig.6). For the free point alone or a pair of points connected by a spring, obliquity is fine ; but not fine in all other cases.

fig.6: a) simple, but it escapes from linearity: busted! b) the spring problem in general: busted !
a) b) [ NOT READY!! ]
c) a similar one... 2 pendula connected by a pring: busted ! d) wheel rotating with full grip, driven by spring wich can become obliquous: busted !
c) figure c) missing. d) figure d) missing.
e) points sliding on circular support, interconnected by straight springs: busted ! f) rigid body plus spring: busted !
e) f) figure f) missing.

it's little stuff, but let's be precize.

All linear dynamical systems are expressed by the seme, canonic, system of linear, ordinary differential equations. Let's give the expression of that canonic form, and go quickly on. The expression is:

A*qdd + B*qd = - D*q + k

where A,B,D are N_DOFxN_DOF matrices, and qdd, qd, q and k are vectors featuring N_DOF elements. So simple: let' do not waste much time on discussing the detailes... they are pretty simple and even if not all, or straight none of them, are clear to You diear Reader, just don't worry. The discussion to come is pretty well understandible even wothout those detailes.

That set of fake linear dynamical systems, anyway have their importance. I invite You to explore the topic on the lane of a few suggstions I make here. Most of You, dear Readers, probably are young, and most of You probably like some kinds of videogames. Now. A very exciting 'excercice' - so to speak could be this: try implementing, in JavaScript possibly, a little interactive simulator which can simulate any true and any fake linear dynamical system. I'll make an effort to make available a ready implementation of it, done by myself. Anyway, if You can do it Yourself, is not wasted time... better of course if done in group. Now. You will notice that the distinction of the 'external', usually interactively given forces, sum up to forces coming out of the potential function which may include the contribution of classical gravity and springs. In those fully linear dynamical systems, the concept of potential-energy of calassical gravity and springs displayed legally, schrinks in the D matrix. That's nice and good and everything, but we can push that forward, to include, in a numerical simulation only, clearly, to be able to handel also the 'fake' linear dynamical systems: by dropping the D matrix, and obtaining the vector resulting from the product D*q in other ways than symbolically. Let's start playing around with these fake linear systems to get an idea on how to go about this.
Playing around with the fake linear-systems, is useful in fact, because we see that dealing with the forces excerted by springs, is best done by numerically doing the derivatives of the potential-function with respect to each of the generalized coordinates which feature in the parametrization we have chosen to do, of the mechanism at hand. The important thing to notice is that given the springs, the potential-function can be procedurally 'built' up as the symbolic expression for a single sping's potential is simple and always the seme... . This is very useful to know. Usually, springs don't have all that internal friction, so, calculating that and adding it to produce dumping, is not so important and would be also more tedious to do. But adding friction acting along the supports to which the various material-points or 2D rigid-bodies are tied, is already done by one of the matrices in the linear systems' differential-equation-system! - did You notice it? It's the B matrix. A well-structured little simulator of this kind, can then be adapted to simulate any linear and fake-linear system easily... it's thus, by itself, once done, also a true little simulation-engine which can be used as a library to construct various little videogames which features the interactive, real-time simulation of any simple linear or fake-linear mechanism. The parameters of the matrices can be obtained by hand-calculation, or with a symbolic calculation program like wxMaxima, and the values of the matrices of the system of -linear- differential equations, can comfortably be plugged into the sourcecode of the given program being made, and that's it. Why this digression? - because we'll see that with non-linear systems instead, symbolic expressions are practically hopeless except the case of very simple NL-mechanisms... that's the reason why all the idea of numerical simulation has to be reconsidered and reduced to fully procedural numerical differentiation for everything, from the parametrization to the Kinetic Energy function, potential energy function and the whole Lagrangian Equations of Motion: as it's done in the LDNFS simulator which I deviced myself. It works only for mechanical systems in it's original form, but it's a good lesson for any simulation-program specialized in the simulation of any, typically non-linear dynamical system. It's more complicated than a typical simulator specialized in the simulation of generic electronic circuits, but it also does some more dirty job. If curious, or if the LDNFS simulator stuff is too much for You, then read comfortably a describtion of how a circuit-simulator such as SPICE typically works. You can always go back to the LDNFS simulator's functioning.

The essence of a Linear Dynamical System is that once we have how many parameters are needed to define it's state, it's configuration in an instant, call this number n, well they all lead to a system of n, 2nd order, ordinary, linear differential equations. Which can be solved either analytically comparatively simply, either numerically, with the additional benefit of being able to esteem how precize our numerical solution, our numerical simulation of the system's evolution in time, has to be in order to give enough faithful results.
Some terminology: ordinary means non-partial... it means that all will depend, at the end, on 1 variable: the time-variable, call it now t. Usually there's the time-factor in all dynamical systems, since they vary they state in time most of the times: so a dependence of the final result on time, is pretty typical. However, there are cases where there are other variables in play, such as the position's x,y,z coordinates if it's a field, to say where we want to evaluate the field's force-vector at a given time t. Or, a simpler example is the x-coordinate of a vibrating string whose shape-curve's y coordinate we want to see at a given time t, etc.

An important note about Linear Dynamical Systems is that even thought they are the simples Dynamical Systems in general, there are different definitions of it... and it's important to understand that all state the seme thing. The output-signal of a linear dynamical system has the seme frequency, but it's just rescaled, or left as is... remember, no amplification if allowed for now.
Another way to describe them is that if we input 2 different stimuli in the system, say a(t) and b(t), and the system outputs to a(t) oa(t) and to b(t) ob(t) ; then to the input a(t) + b(t), it will output ( oa(t) + ob(t) ).
Yet another way to describe them is if the system is input a single short square-signal or an approxinamtion of the so-called 'dirac-delta' signal, the system output signal which is the sum of: either stable sinusoidals (infinitesimally attenuated to the scale of how long we record the output signal to then study it), or exponentially decaying sinusoidals ( in other words: the amplitude of the sinusoidal decays exponentially ), or an exponentially decaying signal ; theoretically also exponentially growing sinusoidals or plain exponentially-growing singnals are possible, but that happens only in the case of proper input-signals. Doubts will dissolve. See some examples... . I give one example: it seems bad, but it's actually not so bad.

[ line 1 : example function, solution of a linear diff-eq system] ;
[ line 2 : example function, solution of a linear diff-eq system] ;

They can have different real-world implementations: be careful! Mechanical, Electrical, or Electro-mechanical.
For Mechanical, and electro-mechanical systems and with some creativity also electrical systems, the final 'resulting output' signal is given after the system has been schematized with the Lagrangian Dynamics, schematization which leads to the Lagrangian Equation of Motion ( or system of them ), which is written explicitely so:

Expanded for of the Lagrangian Equaion of Motion

with i = 1,2,3... according to the number of DOFs which the dynamical system to simulate has ; here i = j obviously, as to arrive at the solution, as many equations are needed as many unknowns there are. Where if i = 1 , yelds specifically that j = 1 too, and that the dynamcal system at hand has 1 DOF. Systems with 0 DOF can't move whatsoever, in fact.

Don't worry if it's not clear... . It's usage is completely mechanical, and a basic introduction to it can clarify doubts. After reading the entirety of this essay, familirizing with that base-approach and implement simulations with it, will be amazingly easy and fascinating.
For electrical circuits instead, they use also another schematization method which is called Modified Nodal Analysis ( MNA for short ).

It's time to say that any combination of interacting ( weakely interacting ! ) linear dynamical systems, gives again a linear dynamical system. In electronic circuits, the interfacing is done either by forcing a node of the circuit ( that's the input ) to have a certain voltage at each time according to a certain signal we want to input into the system as time goes on, and the output is measuring the voltage of any of the other remaining nodes. The mechanical equivalent of this, is attaching an actuator to a mass-point of a mass-spring-pulley system. Inputs can happen on more nodes (if it's a circuit) or more mass-points (if it's a mechanica system) : detailes on this will be outlined a bit better later.
The most subtle fact which allows us to literally bust that the system is linear, is that it's responce to the interaction, does not really depend on the current state of the system, whatsoever! If we raise the strength of the imputs, the final result is just amplified... . How to say it? - the system lacks surprizes!
The mechanical examples are more powerful here: if we remove springs and all potentials, the system's parameters go on changing totally at the seme rate! The system is thus limited to just distribute the effect of an imput, and have a 'memory' of it. But it's a set of as many numerical values as many DOFs the system has!

And well, there's another method to interact, but it's presentation will only corroborate out previous statements' correctness. Let's see it.

Changing velocity instantly to a mass-point of a mass-spring-pulley system ; or it's electrical equivalent in case of interfacing with an linear electronic circuit: imputting a delta-dirac signal into a specific node of the circuit. Both cases share the risk-factor of potentially breaking, damaging the physical implementation of the system. In the mechanical variant, collisions are needed, wich can damange an alement. In the electronic variant, hight-currents are involved, and these can burn resistors, wires, etc, and can mechanically deform things due to high-force electric fields and magnetic fields.
These are problems of implementing the ability to transfer to the system, it's initial configuration.

Ones this is done, watching in time the evolution of any linear dynamical system is the seme stuff... see some examples... they mey be deceptful as many effects sum up, but that's just summation... . We need to be enough sharp to notice that we are fare from anything really so increadibly complex in it's behaviour.

Linear Dynamical systems are important because they are easy to study, to simulate, to predict their behaviour, to simulate them numerically ; and thus they are also easier to design to behave in a desired manner.
They are important because they are simple.
But it isn't only them who exist out there... so, let's progress further.

Why are also Non-Linear Dynamical Systems important?

The more basic reasons

The most obvious reason is: because they are the majority. There are many of them in general, and many types, of, Non-Linear Dynamical Systems. That's true ; however, there are worse ones and less bad ones.
The less bad ones are those which can be literally plugged into a numerical simulation routine, as if it were a linear dynamical system, and the simulation will, very likelily, be faithful.
As a start, let's say that Mechanics gives, practically always, not-so-bad non-linear systems. The essence of this consists in the fact that the second-order derivative in the final diff-eq system is always at power 1... so that it's non a crytical non-linearity. More on this is treated in my book on the L.D.N.F.S simulator where I justify this fact in detail. But let's go on... You can turn back to the detailes later, whenever comfortable.
So, that's one of the reasons why non-linear systems featuring in Mecnanics must not be overlooked: as they give as a first, not too high stair-block to overcome in order to manage finally, to overcome the other stair-blocks and familiarize a bit with non-linear dynamical systems in general. Les't see some examples.

fig.7: A classic example of a non-linear dynamical system that can be simulated with Lagrangian Dynamics:
I chose to parametrize its configuration with the angle of each of the 2 wheels.
It's actually a bit tricky to get it right, because:
one has to consider how much the internal wheel has managed to roll according to it's angle (angle 2), on the wall of the outer wheel.
Sorry, your browser does not support inline SVG.
The vector of generalized coordinates ( the 2 angles in this parametrization, expressed in radians ) is:
[ 3.1 , 2.4 ]

Another classic example is the double-pendulum (fig.8).
We'll come back to it.
What's the general form of an NLD-system?
A*qdd + B*qd + C*qd = - D*q + k
where the form, rigorously, is:
A( q )*qdd + B*qd + C( q, qd )*qd = - D*q + k
where the dependences have been indicated explicitely. The C matrix multiplies in fact still the qd vector, but the dependences are more complex... C is not made of constants as B is. Also A has changed: it has more complex dependences... effects are not just redistributed always in the seme way, but the way they are redistributed, depends on the Mechanics' regime's Dynamical System's configurations at each instant. C is the worst of them, but that's it. We know this, and let's don't worry about it. The new system of diff-equations can be plugged in a numerical simulation procedure as if it were linear as it's previous variant illustrated in the previous section.

We have introduced the NLD-systems offered by the Mechanics' regime. Let's give a closer look to them.
We start trying to catalogize them: hard, but we have 2 extremely useful candidates for a start. Rememeber that we are talking about types, categories of NL-dynamical sysems, not single systems... at most, we can use a specific system to represent a category.
One, is the worst 1-DOF dynamical system from Mechanics. It's the piston... at laest it does it's job pretty well.

[FIG.? missing]

The worst 2-DOF system, well, it does not really make all that sense to search for it: because of the difficulty in finding the worst combination of 2 joint-types. Use Your imagination here.

Than, it's just a matter of reasoning on it a bit, to arrive at the second of our candidates: which are the simplest types of NL dynamical systems against the number of DOF's?

The less bad 1 DOF system is? - point tied to a parabolic curve.
The less bad 2 DOF system is? - point tied to a parabolic surface ; or the polynomial approximation of a double-pendulum (!) .
The less bad 3 DOF system is? - orientation of a 3D rigid-body according to my first theorem on Numerical Methods: let's take it directly from the LDNFS book!

R( q0, q1, q2 ) = Id + E + E2 =

1 0 0
0 1 0
0 0 1
0 -q2 q1
q2 0 -q0
-q1 q0 0
-q22 -q12 q0*q1 q0*q2
q0*q1 -q22 -q02 q1*q2
q0*q2 q1*q2 -q12 -q02

Like the visualization of matrices with HTML tables? Not bad, eh?

As I stated in the introduction, I trust in the intelligence, creativity, curiosity, and whatnot, of my Readers: so I assume that You realized that I've just listed 3 of the simplest non-linear-dynamical systems but which feature only stong inner relationships. The point constrainded o a piece of parabolic frame which itself can slide along a straight rod, does not feature as the it's cheating since one parameter, how far it has slid on the rod, is decoupled, it's totally independent of the rest. I hope this suffices to outline what kind of examples I was inviting You, dear Reader, to look for.

Now. We are close to it... . Let's state it... : a crucial point about Non-Linear Dynamical Systems is that often, they cannot be approximated with a similar linear dynamical system, whatsoever. And this happens notwithstanding the fact that all non-linear systems have a linear core, but this linear core often contributes so little to the final dynamics of the system besides giving a scheme, an existance to it, that it's upsetting. But what about polynomial systems? - Yes! They do it, and it's discussed in the LDNFS book.
But meanwhile hay... Non-Linear Dynamical Systems have forced us to discover and catalogize Polynomial Dynamical Systems: either when they are as they are, or when chaining of them is needed... is that little thing?

It's upsetting at first glance, but an evidence as soon as we start considering it, is moreover that Linear Dynamical Systems' resulting systems of differental equations are limited to featuring only summations and subtractions! Never a product! It shouldn't then even be so much upsetting, if not for the final complexity of the bahaviour of a non-linear dynamical system, is that Linear Dynamical Systems are not the final word as far as Dynamical Systems are concerned. More is needed, in fact.

In short, Non Linear Dynamical Systems spontaneously push modern approach and research, to a deeper understanding of Numerical Methods, as analytical solutions are hopeless. And they push also to be not just good, but to be, to the least excellently good with Linear Dynamical Systems: as they are only a starting-point toward a deeper understanding of more complex things.
This pushes also to a deeper understanding of the concept of differential equation and system of them, and to roam across a wider variety of them. Concentrating on a better understanding of the undelaying concept, rather than the consideration of it in the optics of an expectation of an analytical solution to them.

So, Linear Dynamical Systems never feature products in the system of differental equations describing their motion ( which it does by giving how the generalized coordinates with which the system's geometry was parametrized, do vary in time ). While Non-Linear Dynamical Systems do feature also products ! And products can easily give place to the presence of quantities which span orders of magnitude... as orders of magnitude are obtained by products. And what does this fact of the orders of magnitude mean concretely? - Well, it means the we can have very quickly-happening changes, as well as very slight changes. Both equally difficult to frame, as one lasts an infinitesimal time, the other instead features infinitesimal quantities.

As already mentioned before, it's fine that Non-Linear Dynamical Systems have a linear core but the importance of understanding these systems consists also in recognizing that that addition can contribute so much to the final nature of a Non-Linear Dynamical System that it must be taken into account!

Non-Linear Dynamical Systems let us meet systems which have strong inner relastionships, and have much more many inner relationships overall, than do Linear Dynamical Systems.

They are the general case, and so sharpen our catalogization-tools. But at the seme time, they also invite us to trust our categories and catalogization-methods to the right extent, and not beyond. This is because Non-Linear Dynamical Systems are more authentic! Yes! And therefore require considerable care in the attempt to categorize them ; as this attempt can be difficult, and may also fail at all. And why is this categorization difficult? - well, because there are in fact many types of non-linearity. Where we find different levels, different strengths of non-linearity.

Non-Linear Dynamical Systems can do many useful things which the linear cannot whatsoever. To mention some, let's start from the Mechanics: the piston, which transforms a press/pull sequence into torque-force excerted on a rotatig disk... the essence of transforming thermal or thermo-chemical energy in mechanic energy ; or a typical steering-mechanism with ammortized wheels, which is essential for the automobile. And now pass to the electronics: diodes, transistors, Operational-Amplifiers, active filters, PLLs, bistable systems such as flip-flop memories or similar circuits, and Sample & Hold circuits. Let's add also some examples from the physics and chemistry in general: magnetization processes, most of thermal machines, most of electro-chemical and magneto-chemical systems ( solid-state or not ), and many others.
They can, in fact, do complex things... as they are not the usual passive systems. Reason why at the basis of most of modern technology, lays the strategid use of various Non-Linear Dynamical Systems.

They bring us the - practically explicit - invite to consider carefully high-efficiency, hight-energy, and cascade-like phenomena ; as well as things which seem complex, but which are not so complex, after all, and are also very useful in application. Since these types of phenomena are very common they show us that Non-Linear Dynamical Systems are also very wide-bridging as to the fileds of application or fields of research they arise in... . As stated in the previous paragraph, think to: most mechanical and electronic systems, most phycisal and chemical systems.
So, it's important to familiarize with them: they come in handy a bit everywhere, and they are often a must to understand lots of things, and understand them well. Or understand them well and utilize them well if it's a piece of technology.

The more subtle reasons

Some friendship with Non-Linear Dynamical Systems is important, since there are so many of them in real-life, there are so many types of them, and many of them can do magnificient things. Even though sharing some common features, bring us to be more wise. They bring us to have a sharper look at the world around us, at the world we live in, and at ourselves.

Their understanding, the deeper the better, brings us to be more wise. As well as more modest and careful. Although they, at the seme time allow us also to re-consider our resources as human beings. Some sorts of very generic, self-referential, non-linear systems: not dynamical systems of course ( as we had cast out this for any living organism whatsoever, alredy in the previous section ). But something certainly self-referential, and non-linear for sure. And this is important to understand and see as a fact, as it's a very generic scheme which, although generic, at least does not put forward excessively hazardous assumptions and statements which could not be verified and thus would most likelily be simply wrong in general. Favouring the wasting of our potential resources due to a massively wrong, let's say superstition-based, conception of ourselves as human beings, our real potential resources, and, say our place in the Universe (or how to call it?).
A a correct understanding of ourselves as human beings and our real potential resources, which is important because science is a human activity. Any way one tries to see it, after all, it's men who do the science in fact ; and it should not be then even so surprizing if in the view of modern science, we had a sharper and deeper sight also on ourselves as men, our resources in terms of human qualities and values; in short, our place in the Universe.

Some understanding, the deeper the better, of Non-Linear Dynamical Systems, opens up, it widens and sharpens the way we can think of the world, live in the world, and be constituting part of it. Thanks to at least a minimal familiarity with them, we simply become moe far-reaching in dealing with self-referential systems... and this in general! - we ourselves as human being are self-referential ( think to the fact of being self-conscious, autonomous, and having the faculty of free-will: about being autonomous, we will discuss later ).
Some understanding, the deeper the better, of Non-Linear Dynamical Systems, allows us to break out of limiting and way obsolete ideas such as that a Dynamical System can be described only by a system of differential equations. Let's see what else we have here:

  1. there are event-based dynamical systems: the simulation of the idealized rolling and collision of biiliard-balls on a billiar-table, is one of these. It's not the final word in event-based dynamical systems, where things go on in continuous time but also 'instant' event happen, but a solid example. Another example if the simulation of any mechanism which simly includes springs which excert dumping-force algong the direction they are lied, but linear instead of viscous. Since linear dumping is expressed by a non-algebraic expression, it can only produce a numerical solution, but not an analytical one. As a 'modulus' operator is used, which is even-based... one thing when the parameter is positive, another when negative. An expression line |x| is not even derivable in fact, thus being cast out as candidate to figure in any legal differential equation. But somethig perfectly legal and expectable in the context of a numerical simulation. In which the passage from negative to positive x in the evaluation of |x|, is simply an event.
    Another candidate, even if less eligible, may be the real-time simulation of the magnetization of a piece of ferromagnet: anyway, just a dandidate... I don't have sufficiently strong background to work it out fully and justify the final consclusion fully. An idealized simulation of any digital electronic circuit is event-based, but that's purely event-based, even if the 'simulation' itself can be made to seem continuous and happening as the reaction of a dynamical system to external stimuli, is real-time.
  2. there are dynamical systems based on maximum-flux. Thermodynamical machines and their simulation? - no more detailes on this. Please consult extrnal meterial, if interested in the topic.
  3. mixed systems: NL electo-mechanical systems, mechanical systems on which forces are excerted by thermochemical phenomena, etc.
  4. all Fluidodynamics, or call it whatever way... it includes aerodynamics too, and whatnot.
  5. and so on... .

From this we see many more applications and manifestations of the Principle of Least Action. And we see how minima and maxima are facets of the seme medal. As well as the fact that failing to understand perfectly the theory of differential equations ( and, most importantly systems of them ), is not all that big deal.

Let's go back a little to the topic of self-referentiality.
With Non-Linear Dynamical Systems, wee meet not just simple ( and weak ) self-refenrentality, but also various degrees of it. Degrees of which can very well span orders of magnitude. And we are there... back to the inner relationships. Studying and simulating Non-Linear Dynamical Systems teaches us also that inner relationships and self-referentiality are 2 facets of the seme medal: a very precious one indeed... . A single Non-Linear Dynamical System can encapsulate inner relationships, or now let's call them also an inner network of self-referential relations, whose strengths can be anything between the very weak and the extremelay strong ; where the very numerosity of the relations can be large as well.

Let's come back for an instant to the story of us being autonomous: well, an understanding of, and friendship with, Non-Linear Dynamical Systems very likelily makes us wiser, as stated before: but thus makes us also more autonomous! People with more resources available, a fairly correct recon of these, and a much higher ability to employ these resources in an optimal way.
Seeing moreover the parallelism between different systems, invites us utterly to learn to optimize better: this makes a lot of sinergy with the fact of becoming more autonomous, as we would be driven at double-force to manage optimally our resources.
Well, we are autonomous also inthat we understand, acquire, learn also from the context.
And why did I say that an understanding of, and friendship with, Non-Linear Dynamical Systems very likelily will not only make us wiser and thus more autonomous, but also people with more resources? From where would that surplus of resources come? Well, from a realization that it's easy to make after having familiarized with NL-systems: that we as human beings acquire, learn and understand, also from the context and not only from what is explicitely said. Why do I say this now? Well... I program computers: I rely on them doing all and only what is told them to do by the sourcecode of a computer-program. And so far we should have become aware of the essence of the relationship between humans and computers: they have give us resources which are practically complementar to our resources as humans. One is infallible and extremely fast in performing a series of arithmetic, logical and data-storage operations ; while we have the feature of being self-conscious, the ability to drain information also from the context of rven out of wholes in an information which is supposed to be complete but which is not, the ability to reason, to imagine, and to create, to discover, to self-enhance and adapt, and the faculty of free-will.

As I stated before, NL-systems are more authentic: do you see now why? - because while linear systems are passive inthat they don't output anything "of their own", NL-systems do. And this teaches us to realize how important it is for us as human beings, to be authentic, and how to do this, and in what this may consist: well, to the least it consists in putting into things something of our own. Well beyond the mere personal interpretation. And this is important, for good resources of single paople, it is important not just to exsist for their own sake, but also that they be canalized into a positive, constructive, productive collective human activity... at the basis of a better Civilization. Better in everythig.

As last thing, let's stress one more point about Non-Linear Dynamical Systems. Let's start by stating that probably You have realized that already in the first part of the discussion, some types of words appeared which, up to this date, seldom or never appear in scientific communication and textbooks. Words which point to human characteristics, human behaviours, human actions, and human social interaction. Well, at this point the reason of this and the danger in limiting scientific narrations's lexic too much, should be evident. Because indeed, as You probably have realized, discussing non-linear systems utterly invites to acquire into scientific lexic new words which were not needed in the old science. Let's list a couple of them.

Adjectives :
authentic, amazing, colorful, interesting ; receptive, rejective ; passive, active ; sensitive, sharp, subtle ; energic ; smart, dumb ; sharp, single-minded ; efficient ; fast, slow ; responsive, irresponsive ; strong, weak ; fragile, rigid, delicate ; precize ; accurate ; complicated, complex, simple ; explosive, impetuous ; invasive, non-invasive ; reactive, adaptable, versatile ; prevedible, imprevedible ; one-sided, limited, poly-sided, colorful ; coherent, consequential ; difficult to frame, anti-modern, modern, modernized ; easy, easy to deal with; deep, shallow, deep-digging, superficial ; self-referential ; self-organized ; destructive, constructive ; volatile, non-volatile ; polyvalent, multifunctional ; superficial, deep-digging, deep-reaching, far-reaching ; decepful ; extreme ; external, inner ; paradoxical, controverse .

Substantives and Verbs:
indentity ; reaction, response ; relation ; relationship ; friend, alleate, enemy ; communication, information ; welcoming of something ; rejection of something ; acquisition, assimilation, storage, elaboration ; adaptness to something ; authenticity, uniqueness ; putting something of one's own ; ability to surprize ; character ; living ; self-regulation ; evolution, transition ; one's history, one's past ; happening, event ; network ; bias, orientation ; state, change of state, updating someone of one's chage of state, communicating one's state of change, someone delivering news ; to inculcate something in someone ; monotony ; friendship ; familiarity ; arousal ; jump-like bettering ; harmony ; synergy ; self-limiting shell, living in a self-limiting shell ; extreme reaction ; heartbreak ; to breeth something too deeply into one's lungs, to take something easy ; to suddently realize something ; for something to be impressive ; something pointing to itself ; making a coordinated effort ; multiplication of effects ; hightly bread faculty, latent faculty, latent talent ; coordinated effort, collective effort ; cooperation, partecipation ; reaction-time ; retroaction ; turning-point ; to misinterpret someone, to misconceive someone, to exchange something for something else ; event, happening ; to endure something ; to stand one's position ; flavour, type ; dimension ; transmission, delivery, to inform someone of something ; to steer something, to drive something, to be briven to do something, to be steered towards something ; to achieve something ; intrinsic feature ;

This casts light on the fact that in order for one to have a modern and efficient attitude toward a scientific activity in the era of the Third Millenium, as long as one chooses to go for that kind of activity, there's not just the need to have some friendship with NLD-systems, but also the utter need to acquire familarity with a scientific communication which be adequate to the modern era in which it was born and exists. I mean that in an efficient, precize and easily understandible scientific communication and teaching, the afore-listed words and many others, must be used: and their correct usage must be mastered in order to avoid that the communication, or direct teaching, fail completely. Wherease it's important that it does not fail, othwervise the next generation will undergo a formation which drives them to become anti-modern scientists, and more in general, people with an anti-modern attitude which has a higth risk to make damage to themselves... and maybe also the environment they live in, and... or... the environment they shape for themselves to live in. Think to the pollution of environment, think to the evident and strong, plus widespread, tendence to understimate the importance of an advancedly functional pedagogy, tendence to understimate the importance of the spreading of up-to-date scientific knowledge, and so on.
The Theory of Relativity ( I mean the simpler one, the one called "special" ) is the key for understanding Electromagnetism.
And well, from all that this essay discussed until this point, it should not be too strainge that:
the key to understand modern science, as well as to understand modern technology, and to achieve having a satisfactory life in the modern age, the key is, I don't say understanding, but, having made at least a minimal friendship with NLD-systems.
There's not much to say left, as it's our modern age's generations' masters that we have to listen to: NLD-systems. I try to paraphrase a bit also their implicit teaching.

Basicly it's a matter of abandoning a view of what is historically called the world of "hard sciences" as a counterpart, an extreme opposite, of the world of "living sciences": as this view and attitude toward the whole field of applied experimental excact-sciences, has become completely obsolete. It underwent total obsolescence because as a metter of fact, that attitude would simply preclude the ability to understand and treat correctly most of modern science. Most of it! Here I limited the examples on classical mechanics and electronics, on purpose... to show that those once which used to be most etiquetted as the representant of the "study of dead matter, dead objects: mechanisms and electronic circuits" are those wich need to be reconsidered first... let alone the rest! But once these are viewed and treated in a modernized way such as the one I tried to gently guide the Reader to, the rest comes practically by itself... . And it won't even be all that difficult. Every notion will self-settle in the right place and form a coherent, somart, easy-to-travel-through network of interrelated facts, and laws, parallelisms, similarities, utter differences, and so on: a living network (!) ... that's why! A network living in our heads, and providing us with a universal, modern, new wisdom.

Maybe I am self-repeating a bit, but let's formulate one last thing in which the aforementioned new wisdom will make us better. We will gain a more scientific, a better, a deeper understanding even of the plain cause-and-effect. Maybe it does not seem all that big kick at first glance, but it is! - [...]


In scientific communication text, figures, schemes, tables, formulas are used simultaneously. They are co-present.

Now, the text part expresses thought: it guides the reader's reasoning trying to guide him to understand the base-logic of something. But that's not enough for modern science... in most of the cases there isn't a base-logic in fact. To the utmost there's an approach.

But approach is too little because detailes collapse into a too global view: also the detailes must be presented sooner or later. But they are interconnected by a subtle network or logical relationships. And thus, and here we are, the task of guiding the reader to gather an insight into this network getting both global and close-up view and knowledge of it, is crucial.

It's true that modern science is more complex, it's true that Non-Linear Dynamical Systems are more complex, but not so complex. They can be mastered and need to be mastered by a modern-era's man.

Not taking enough care of writing the text part well, the explanatory part of a scientific text well, is a tremendous mistake. At this point of this little chapter, this should be obvious. It's mere carelessness or incompetence. Wherease carelessness can be enough to trigger the creation of very bad texts also by people who are enough competent. While the other way it doesn't even go: incompetence in unadjustable... but not a problem itself as long as those who undertake the delicate task of writing a study-book, is as competent as possible.

The expanatory part in a modern scientific text is as important as the presentation of the right formulas, and the presentation of clear and suggestive figure or scheme wherever it's needed or even just... simply useful.

A conrete example, to conclude, is the attempt to describe how a single LED can not act as a photovoltaic panel because it can not take so much light in it to supass the treshold-voltage beyond which it passes to the conducting (in this case current-generating) regime. But if we put many many such LEDS in series and illuminate them all with a reasonably strong light, it starts generating and all LEDs pass in the conducting regime. That is a photovoltaic panel, just not miniaturized... . Apparently this could be dismissed so. By putting 1000 miniaturized LEDs in series, if each LED generates an, say 0.002 [volt], tension between it's 2 poles, then as there are 1000 of them, well, they multiply up to dive 2 [volts]. Yes, but there's much more to say about this, and if a modern-era scientific author does not have an adequate writing-capacity, he is tempted to dismiss further aspects which are more subtle... so guiding the reader to pass by the more subtle aspects. Where, in the optics of Non-Linear Dynamical Systems, the subtle aspects can surprize... . So it's utterly important to describe them too. Then the technical detailes can be laid down and they find their right place in the global framework. In the case of the diode, it will be evident that the key to understand why it works, is to describe to the reader, guide his reasoning ti see that our aforementioned group of 1000 LEDs work in generating current because they literally manage to make a collective effort is summing up the very tiny voltage-contribution which each single one puts into the global stuff: they cooperate. And togather manage to convect each's very tiny contribution into one single channel of effect... the power-line in which we have inserted the series of thos 1000 diodes. How they do it and excactly how this all happens, are detailes of modern physical chemistry... but detailes are not so hard to lay down. But The aspect which was harder but nesessary to convey in a modern worldìs scientific explanation, is the fact that the series of 1000 diodes form togather a tremendously complex Non-Linear Dynamical System... one which is extremely useful. Solar panels are extremely important in fact: they don't last forever, they are hard to manufacture, that's true... but they are light and portable... and thus allow for example artificial satellites and space-telescopes to have them and get enough power for their functioning.

How to bust Non-Linear Dynamical Systems ?

At the dawn of the 3rd millenium in the standard historical time-classification, it can be crucial understanding Non-Linear Dynamical Systems, to live in harmony with them. In this short essay, I have tried, so far, to outline the reason why this is a universally true fact. So, at this point it's perfectly in place to dedicate a small chapter which outlines how to bust, when a system is non-linear: case in which it's advisable to raise the level of vigilance. Let's don't go straight... go non-linear instead!

How to bust Non-Linear Dynamical Systems, and admire them, and play with them, to make our life-experience richer and more colorful?

Anything which has very restricted optimal-functioning regimes: diesel engines are like that, for example. I left this example on purpose for this last section of this little essay, because it is a trivalent example in which the dynamical system at hand is mechanical ( the piston plus transmission-axis mechanism in the engine ), thermal, and chemical ; simultaneously.

Epilogue: Thank You for Your attention... it's been worth it !

One day I came back from the town, in particular the near-forse suburbs of it, and I was received with:
-"Ah, I almost forgot to ask... have You seen anything interesting?".
And I even did't have the time to think over what I was answering when I was already pronouncing the answer.
-"I've seen a flowering Nature, in which a rotten civilisation is sitting, which also damages that Nature: a damage whose signs are evident at this point... . Damages which however, still allow to forsee a Good Tomorrow.".

Well, a final, really interesting fact I have left as a surprize about NL-systems is that an adequate familiarity with them, sharpens greately our insight into the world of inanimated objects: but so it will also invite us to re-consider their potential complexity, and complexity of their behaviour. And this will intead loosen a bit the now-obsolete sharp distingtion between the world of the living and the inanimated: by assimetrically narrowing the world of the unanimated to that of the animated. Thus making us able to live in harmony with our environment which we will then be able to shape better at our turn to our needs, and more in harmony with each other as human beings.
To support this, I tell an amazing episode that happened to me and struck me heavily. I was around 16 or so, and in a class, our physics teacher had introduced us, little before, to some suprizing properties of crystals and the evolution of the idea of the difference between the living and the unanimated. And in that class, he showed us a footage of a bunch of simple experiments with crystals.
And well, in the footage at some point a piece of perfect crystal of size 1-2 centimeter per side was being held by a man: he put it down on a table so that it was standing there still, and hit it heavily with a large hammer: nothing happened... the happer rebounced as if he had hit a gummy surface. Than he took a small hammer, took the took the crystal, held it wi a special angle easing one of the crystal's vetices in the table's surface, and hit gently another upward-pointing vertex of it with the hammer: and the crystal fell instantly into pieces, and a good part of it literally pulverized... the white-ish powder was evidente. And the teacher said:
-"Did You see ? He killed it... . He killed the crystal.".
It took my breadth, literally. I realized at that point that the whole idea of science, technology and ourselves as human beings, our relationship with Nature, science and technology, and the way we realte to each other, had to be re-considered competely: as in the modern world a completely new, and a much more advanced and potent wisdom was needed. Part of this new wisdom is a heavy exposure and at least a descreet familiarity with Non-Linear Dyanmical Systems. Coming back to the crystal thing, well, what should come to our realization in a modern optics, is that the crystal, in one case acted as the most perfect implementation of an elastic collision between the hammer's head and the crystal's own surface, while in the other case the most perfect iplementation of the unelastic collision. Such far removed opposites in behavious, performed by the seme entity. Well, that entity, what can we expect to be, if not a heavily non-linear system?
At this point we can state fairly safely that the complexity of the behaviour of some Non-Linear Dynamical Systems is similar to the complexity and adaptability, versatility of our actions as human beings... this narrowing the world of physical study-objects to ourselves: so guiding us to abandone the now overcome idea that excact experimental sciences are just a world of cold facts. What the crystal was doing in the 2 experiments, is in fact assimilable to 2 distinct human bahaviours: one is the total rejection of something, the other is instead extreme receptivity to something ( the crystal broke because it absorbed in a matter of an infinitesimally small timelapse, all the hit of the hammer ).
A barely similar one is the diode. [...]. We see that these devices as diodes, transistors, S&H units, etc., handle extremely high energies with respect to their size, and do it precisely. Do it maintaining control over all that flow of energy.

Well, it's time to Thank You for Your attention. If You are a boy or young man with balls, and after coming across this essay, You took the effort to come so far in reading it, probably You are an amazing person. And to Thank You for Your dedication put into the following of the lines of this essay, as a gift, I leave You with a couple of experiments, exciting excercices if we want to call them so, which You can bring out with a relatively common and affordable equipment. If You have not been initiated to the amazing world of Non-Linear Dynamical Systems, then tese will initiate You. To a fantastic journey into the world of NL-systems, which will accompany You along the life, hopefully in a positive way: You won't have a hard time finding NL-systems... remember, I also taught You how to bust them!

experiments / excercices

electronics section:
1. take a diode... . with a tension-function generator, make a square-signal with 2.0 volts (or little more) added to it (the signal must go up-down but never go beow 1.0 volt or so), and attach it to one ends of the diode, to the other end attach a resistor, and to that the other pole of coming out from the tension-function generator. Make sure the resistor's value is such that the diode is always active: we are trying to replicate here a sort of current-pump... best if You use one to make sure the diode does not over-heat and burn down. With an oscylloscope then, measure the tension in time between the diode's 2 ends. See how perfectly the diode adapts itself to maintain 0.7 volts of tension between it's poles, as the current which passes throught it varies between 2 values. Raise gradually the square-shape's frequency, and see how far in frequency it keeps working. Then switch to sinusoidal shape. Vary the amplitude of the signal too. Then test how quickly the diode is able to stop the current when inverted. It's extremely fast.
Construct a flip-flop memory with BJT transistors, and see how it works. It would be even more interesting to program a microcontroller to perfor delete- and overwrite operation now putting 0 now 1 into the system... with adjustable frequency... see with the oscylloscope how far the information-acquisition of the flip-flop memory unit can go. Probably it's far beyond the microcontrollers maximum frequency, but it's impressive to see this 'test' with hand-tangible transistors put togather on a breadboard to make a flip-flop memory unit. If too tedious, take an integrated flip-flop unit for the test. That's made with MOSFET transistors of 2 different types combined according to the CMOS transistor-transistor logic, but the concept is similar... also there the data-acquisition speeds are stunning. Those few-cent chips work usually up to 10 MHZ of frequency... well, test it!
3.Test a tension-regulator. Don't contruct it... they sell it integrated. See how stunning the precision of it is, even that of the most basic ones. Consult how it works... You will find transistors and diodes at it's heart: our well-know non-linear friends!
4. Construct a tension-multiplier and see what outputs it produces to various inputs: though be careful with the output... high viltages are likely to come out... rather, simulate it with ngSPICE or another circuit-simulator, so there's no risk of breaking costy oscilloscopes or other lab-devices.

mechanics section:
5. Take a little, working model of the piston with wheel. Attach a spring connecting the piston with a fix point, with respect to which the whole railway is still too. Now rotate the wheel. See how restricted that regime is, where the spring does not worsen it's motility. It will be far more restricted than a linear system's frequency-response. This is tricky though... . In the worst case, do a numerical simulation based on symbolic expressions ( I made it too, use my implementation of the simulation and adapt it ). The idea is to set a gien angular-speed to the wheel ad give a little torque-force to it. Damping must be present both on the wheel both on the piston. To note that the wheel will be maintained moving on, only if the initial angle-speed of the wheel corresponds to the springs's own-frequency.
6. Try an interactive double-pendulum simulation. Particularly important, see how it moves when classical gravity's force is turned down. The motion the double-pendulum will remain formidable. The motion of the single-pendulum will be instead very simple, as the motion-contribution is linear.


by : Simon Hasur (alias 'the Nerd of Algorithms') ;
date: 17 DIC 2016 - 10 JAN 2017 ;
homepage:: www.nerdofalgorithms.altervista.org