Lyapunov function

Lyapunov fractal with the sequence AAAABBB

In mathematics, Lyapunov functions are functions which can be used to prove the stability of a certain fixed point in adynamical system or autonomous differential equation. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions are important to stability theory and control theory. A similar concept appears in the theory of general state space Markov Chains, usually under the name Lyapunov-Foster functions.

Functions which might prove the stability of some equilibrium are called Lyapunov-candidate-functions. There is no general method to construct or find a Lyapunov-candidate-function which proves the stability of an equilibrium, and the inability to find a Lyapunov function is inconclusive with respect to stability, which means, that not finding a Lyapunov function doesn't mean that the system is unstable. For dynamical systems (e.g. physical systems), conservation laws can often be used to construct a Lyapunov-candidate-function.

The basic Lyapunov theorems for autonomous systems which are directly related to Lyapunov (candidate) functions are a useful tool to prove the stability of an equilibrium of an autonomous dynamical system.

One must be aware that the basic Lyapunov Theorems for autonomous systems are a sufficient, but not necessary tool to prove the stability of an equilibrium. Finding a Lyapunov Function for a certain equilibrium might be a matter of luck. Trial and error is the method to apply, when testing Lyapunov-candidate-functions on some equilibrium. As the areas of equal stability often follow lines in 2D, the computer generated images of Lyapunov exponents look nice and are very popular.

RIGHT ANGLE CIRCUITRY

FIG.1 Wrap an AC coil around an iron rod, and you have an inductor. But wrap your coil around an iron RING, and you form a "toroidial inductor."

FIG. 2 A toroidial inductor is interesting because the induced magnetic field remains hidden within the iron core. If the coil was wrapped around the entire core rather than in one spot as shown, then the magnetic field would exist only within the iron core.

FIG. 3 Even if the coil of wire does not touch the core, it still induces a strong magnetic field inside the core. The gap between the coil and the iron ring can be very large, yet this does not reduce the strength of the field within the core.

FIG. 4 Although the magnetic field stays inside, something else does come out of the core. The changing field within the core produces a field of Vector Potential which surrounds the core. This field is commonly called the "A-field."

FIG. 5 We can intercept the A-field by passing a wire through the hole in the iron ring. This produces a voltage at the ends of the wire, and this voltage can operate an ordinary load such as a light bulb.

FIG. 6 Perhaps this figure is more familar to you. The wire which intercepts the lines of "A-field flux" is simply the secondary of a transformer. Note that whenever multiple turns are passed through the hole in the iron ring, the output voltage rises proportionally. Two turns gives 2x the voltage of a single turn.

FIG. 7 We can route the "secondary" wire through another iron ring. It will produce a strong field within that second ring, and if we add a "secondary" to that ring, we'll see an output voltage. It acts like any other transformer, even though there is an extra stage of "A-field" linkage.

FIG. 8 What if we don't use wire? If we simply place the second iron core near the first, then the lines of A-field flux will pass through both and link them together. The result? Nothing! No magnetic field appears in the second core, and the extra secondary does not produce any output voltage as it did in figure seven. WHY?!!! I don't know. I haven't thought deeply enough about this yet...

FIG. 9 The A-field is associated with voltage and electrostatic fields. After all, if we add more turns to a transformer secondary, we get more voltage on the output. Perhaps we can use capacitor plates to intercept the A-field? What will happen? Can we extract energy from the toroid without passing an electric current through the central hole? I don't know.

FIG. 10 In figure seven we formed a strange transformer by passing a conductive ring through two ferrous rings. This idea can be extended to ridiculous lengths. If the iron cores are not lossy (use laminations,) and if the conductive rings are not resistive (use thick copper), then a long chain of alternating rings will transmit energy with little loss and no possibility of electrocution. The rings need not even touch each other. (Just what is Electrical Energy, if it can flow through such a strange transmission line?)

FIG. 11 Figure ten might seem weird, but even a simple transformer is weird in the same way. Stretch the core so that the primary and secondary are far apart. Energy is flowing along the two sides of the core, proceeding from the primary coil to the secondary. Note that the transformer core need not be conductive. It could be made of insulating ferrite.

FIG. 12 With high-mu materials, the transformer core could even take the form of wires. But these wires are nonconductors. They "conduct" waves of magnetic field. The electrical energy is guided by the spin-flipping of electrons in the iron atoms, as opposed to copper wires where energy is guided by flowing electrons.

FIG. 13 Add a SPST switch to the previous "circuit", and we can break the connection between the two halves. A physical switch isn't required: instead we could place a permanent magnet against the wire and cause it to saturate and become magnetically "nonconductive." Note that opening this switch reduces the inductance of the iron ring, and causes the primary to draw an enormous current. BREAKING THE CIRUICT PRODUCES A "SHORT CIRCUIT" EFFECT!

FIG. 14 Now that we've got a wire, lets wind a coil. But what will such a coil produce? A-field! Many turns of ferrous core-wire will give us a higher output voltage, just as many turns of copper wire passing through one turn of iron core gives higher voltage.

FIG. 15 Let's add a core! Barium Titanate should work. Or PZT ceramic (Lead Zirconate Titanate.) Our "coil" should attract such a core, which means we could build a solenoid actuator. Or a motor. Or just use the PZT core to pick up certain things. Things like lint, and little bits of paper. It's not an electromagnet, it's an electro-electret!

"STATIC ELECTRICITY" IS ELECTRICITY WHICH IS STATIC?

"Static electricity" exists whenever there are unequal amounts of positive and negative charged particles present. It doesn't matter whether the region of imbalance is flowing or whether it is still. Only the imbalance is important, not the "staticness." To say otherwise can cause several sorts of confusion.

All solid objects contain vast quantities of positive and negative particles whether the objects are electrified or not. When these quantities are not exactly equal and there is a tiny bit more positive than negative (or vice versa), we say that the object is "electrified" or "charged," and that "static electricity" exists. When the quantities are equal, we say the object is "neutral" or "uncharged." "Charged" and "uncharged" depends on the sum of opposite quantities. Since "static electricity" is actually an imbalance in the quantities of positive and negative, it is wrong to believe that the phenomenon has anything to do with lack of motion, with being "static." In fact, "static electricity" can easily be made to *move* along conductive surfaces. When this happens, it continues to display all it's expected characteristics as it flows, so it does not stop being "static electricity" while it moves along very non-statically! In a high voltage electric circuit, the wires can attract lint, raise hair, etc., even though there is a large current in the wires and all the charges are flowing (and none of the electricity is "static.") And last, when any electric circuit is broken and the charges stop flowing, they do *not* turn into "static electricity" and begin attracting lint, etc. A disconnected wire contains charges which are not moving (they are static,) yet it contains no "static electricity!"
To sort out this craziness, simply remember that "static electricity" is not a quantity of unmoving charged particles, and "static electricity" has nothing to do with unmoving-ness. If you believe that "static" and "current" are opposite types of "electricity," you will forever be hopelessly confused about electricity in general.

ELECTRIC POWER FLOWS FROM GENERATOR TO CONSUMER?

Electric power cannot be made to flow. Power is defined as "flow of energy." Saying that power "flows" is silly. It's as silly as saying that the stuff in a moving river is named "current" rather than named "water." Water is real, water can flow, flows of water are called currents, but we should never make the mistake of believing that water's motion is a type of substance. Talking of "current" which "flows" confuses everyone. The issue with energy is similar. Electrical energy is real, it is sort of like a stuff, and it can flow along. When electric energy flows, the flow is called "electric power." But electric power has no existence of its own. Electric power is the flow rate of another thing; electric power is an energy current. Energy flows, but power never does, just as water flows but "water current" never does.

The above issue affects the concepts behind the units of electrical measurement. Energy can be measured in Joules or Ergs. The rate of flow of energy is called Joules per second. For convenience, we give the name "power" to this Joule/sec rate of flow, and we measure it in terms of Watts. This makes for convenient calculations. Yet Watts have no physical, substance-like existence. The Joule is the fundamental unit, and the Watt is a unit of convenience which means "joule per second."
I believe that it is a good idea to teach only the term "Joule" in early grades, to entirely avoid the "watt" concept. Call power by the proper name "joules per second". Only introduce "watts" years later, when the students feel a need for a convenient way to state the "joules per second" concept. Unfortunately many textbooks do the reverse, they keep the seemingly-complex "Joule" away from the kids, while spreading the "watt" concept far and wide! Later they try to explain that joules are simply watt-seconds! (That's watts TIMES seconds, not watts per second.)
If you aren't quite sure that you understand watt-seconds, stop thinking backwards and think like this: Joules are real, a flow of Joules is measured in Joules per second, and "Watts" should not interfere with these basic ideas.

ELECTROMAGNET COILS USE UP ENERGY TO MAKE MAGNETISM?

Sustaining a magnetic field requires no energy. Coils only require energy to initially create a magnetic field. They also require energy to defeat electrical friction (resistance); to keep the charges from slowing down as they flow in wires. But if the resistance is removed, the magnetic field can exist continuously without any energy input. If electrically frictionless superconductive wire is used, a coil can be momentarily connected to an energy supply to create the field. Afterwards the power supply can be removed and both the current and the magnetic field will continue forever without further energy input. 

ELECTRIC CHARGES ARE INVISIBLE?

Electric charges are easily visible to human eyes, even though their motion is not. "Electricity" is not invisible! Never has been. When you look at a metal wire, you can see the charges of electricity which would flow during electric currents. They are silvery/metallic in color. They give metals their mirrorlike shine. Some metals have other colors as well, brass and copper for instance. Yet in all cases, the "metallic"-looking stuff is the metal's electrons. A dense crowd of electrons looks silvery; "electric fluid" is a silver liquid. And if metals weren't full of movable electrons, they wouldn't look metallic.

During electric currents in metals, the atoms stay still, but the silvery electron-stuff flows slowly along. Unfortunately the human eye cannot see the electric flow. That's part of the reason that "electricity" is so mysterious. Think about it... in an aquarium full of water, you cannot see any water flowing unless there are bubbles or dirt being carried along. And whenever clean water is flowing through a transparent hose, you can't see any flow. Even if the water is flowing very fast, the water-filled hose just looks like an unmoving glass rod. Same with wires: there's no bubbles or dirt being carried along by the electric current, therefore you can't see anything moving. You can see the STUFF that flows, just as you can see the water in an aquarium, but you can't see any flowing stuff.
Even if human eyes could see single electrons, we still couldn't see an electrical flow since the current is extremely slow. Electrons in metals typically flow at a few centimeters per hour, even during high currents. That's slower than the minute hand on a clock! Electric currents OOZE along like silly-putty flowing across a tilted board.
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Seeing imbalances in charge
Here's a separate topic... while the metallic-looking sea of charges in a metal is easily seen, IMBALANCES of charge are not. This get's confusing, since many books call imbalances of charge by the name "charge." They will tell you that charge is invisible, yet they really mean that charge-imbalances are invisible.
Wires contain enormous amounts of movable negative charge in the form of electrons, but they also contain positive charge in the form of protons within the metal atoms. If the number of protons and electrons are equal, don't they cancel out? Doesn't that mean that no charge exists? No. It means that no IMBALANCE of charge exists.
An "uncharged" wire is still full of charge, it still contains positive and negative charge in huge but equal quantities. The word "uncharged" doesn't mean "without charge," instead it means "without charge-imbalance." Yet even if there are more electrons than protons, or fewer electrons than protons, this imbalance is invisible. It's invisible because the greatest difference attainable is incredibly tiny when compared to the amount of charge that's already there. If an object is highly charged; even charged up to millions of volts, the extra charge is like a teacup poured into an ocean. The difference is far too small to be seen.

A "CONDUCTOR" IS A MATERIAL WHICH ALLOWS CHARGE TO PASS THROUGH IT?

The scientist's definition of the word "conductor" is different than the one above, and the one above has problems. For example, a vacuum offers no barrier to flows of electric charges, yet vacuum is an insulator. Vacuum is NOTHING, so how can it act as a barrier to electric current? Also, there is a similar problem with air: electric charges placed into the air can easily move along, yet air is an insulator. Or look at salt water versus oil. Oil is an insulator, while salt water is a conductor, yet neither liquid is able to halt the flow of any charges which are placed into it. How can we straighten out this paradox? Easy: use the proper definition of the word "conductor."

WRONG DEFINITION:
Conductor - a material which allows charges to pass through itself
BETTER:
Conductor - a material which can support an electric current
BEST:
Conductor - a material which contains movable electric charges
Here's an analogy:
Conductor - like a pipe which is already full of water 
Insulator - like a pipe with frozen liquid; a pipe plugged by ice
If we place a Potential Difference across either air or a vacuum, no electric current appears. This is sensible, since there are few movable charges in air or vacuum, so there can be no electric current. If we place a voltage across a piece of metal or across a puddle of salt water, charges will flow and an electric current will appear, since these substances are always full of movable charges, and therefore the "voltage pressure" causes the charges to flow. In metal, the outer electrons of the atoms are not bound upon individual atoms but instead can move through the material, and a voltage can drive these "liquid" electrons along. But if we stick our wires into oil, there will be no electric current, since oil does not contain movable charges.
If we were to inject charges into a vacuum, then we WOULD have electric current in a vacuum. This is how TV picture tubes and vacuum tubes work; electrons are forcibly injected into the empty space by a hot filament. However, think about it for a second: it's no longer a vacuum when it contains a cloud of electrons! Maybe we should change their name to "electron-cloud tubes" rather than "vacuum tubes", since the electron cloud is required before there can be any conductivity in the space between the plates. (But vacuum tubes already have another name, so this would just confuse things. They are called "hollow-state devices." As opposed to "solid state devices?" Nyuk nyuk.)