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October 30

Logistic Map

File:Logistic map examples.gif This image does not seem to depict the Logistic map as it claims. Various features appear wrong, most apparent is that the asymptote appears at x=1 and decreases before the onset of periodic behaviour, whereas we know from File:LogisticMap BifurcationDiagram.png which is correct that the asymptote move upwards from x=0 before the fixed points begin bifurcating. — Preceding unsigned comment added by 109.157.219.5 (talk) 20:43, 30 October 2013 (UTC)[reply]

Sorry,I don't understand what you're saying. What asymptote are you referring to? I can't see anything wrong with it, though admittedly it is hard to follow because it moves from frame to frame so fast. Duoduoduo (talk) 21:19, 30 October 2013 (UTC)[reply]

For small r (prior to the first bifurcation) the long term behaviour of a system governed by the logistic map is to tend towards a specific value x_inf (the asymptotic behaviour of the system). For r=0 x_inf=0, and as r is increased x_inf increases too, until it reaches about x_inf=0.65, and the system begins a series of repeated bifurcations before the onset of chaos. The .gif linked to shows that x_inf remains at x_inf=0 before suddenly jumping to x_inf=1 and then decreasing before the onset of chaos. The two files I linked to clearly do not describe the same behaviour. — Preceding unsigned comment added by 109.157.219.5 (talk) 23:53, 30 October 2013 (UTC)[reply]

I see what you mean. I'll have to think about it tomorrow. Have you tried leaving a message on the talk page of the person who created the .gif? Duoduoduo (talk) 01:11, 31 October 2013 (UTC)[reply]

It seems to me that each graph, displaying an iteration of a logistic function and used as a single frame in the animation, is independently scaled to the frame's height. As long as iterations are monotonic, the asymptote lies on the frame's horizontal edge, but as soon as osciallations appear the highest oscillation peak touches the upper line of a frame. That's why the asymptote seems to decrease: it simply gets lower relative to the maximum value reached.
Watch also the starting point. It is probably constant, i.e. the same for all iteration runs. However in frames 1 through 19 the graph starts at the frames' top, then suddenly jumps to the bottom (frame 20 in yellow, where iteration reaches its limit point in the first step, meaning f(x₀) = x₀) and remains there till the frame 32, and then seems to slowly shift up as iterations span increasing interval (compare animated File:LogisticCobwebChaos.gif). --CiaPan (talk) 19:59, 1 November 2013 (UTC)[reply]

"solving" a circuit for voltages and currents

In any university physics text, you can find statements to the effect that, given a circuit diagram with resistances labeled, you can determine the voltage or the current at any point as a rational function of the input voltage and the resistances. But there is never any proof of existence or uniqueness. I can see how to write down a system of linear equations that describe the situation, but I do not see how to justify the fact that this system will always have a unique solution.

Any suggestions? Thanks. 50.103.239.50 (talk) 22:57, 30 October 2013 (UTC)[reply]

I suspect that they rely on empirical observations to determine if their answer is correct, rather than mathematical theory. That is, they measure the voltage and current in an actual circuit. Indeed, math often can provide extraneous answers. For example, the simple problem of asking what size square room will have an area of 100 gives answers of 10×10 and also (-10)×(-10). Since wall lengths of -10 make no sense, this answer is ignored. StuRat (talk) 23:12, 30 October 2013 (UTC)[reply]

While not a direct proof, Thévenin's theorem shows there is a unique equivalent circuit. --Mark viking (talk) 23:24, 30 October 2013 (UTC)[reply]

StuRat, good point about the extraneous solution. I hadn't thought of that. The reason I'm interested in this is because certain rectangle tiling problems can be analyzed via "circuit" graphs, as in Stein, Mathematics: The Man-Made Universe. (Of course, Stein does not include a proof...) I was hoping to be more rigorous. 50.103.239.50 (talk) 00:04, 31 October 2013 (UTC)[reply]

A single resistor with neither end connected to anything can have its ends at any voltage at all with respect to a reference ground node (though the two voltages are necessarily the same). Any linear network with entirely disjoint parts will have an indeterminate solution. If your network can include reactive components -specifically capacitors- then even a connected network could be indeterminate at d.c. catslash (talk) 00:45, 31 October 2013 (UTC)[reply]

For a general linear network, if the nodal admittance matrix is singular then by Cramer's rule, there won't be a unique solution for the voltages. For the floating resistor described above, the nodal admittance matrix is

{\begin{pmatrix}-{\frac {1}{R}}&{\frac {1}{R}}\\{\frac {1}{R}}&-{\frac {1}{R}}\end{pmatrix}}

(or according to the article, minus that). Even if you have a unique solution for the voltages, you may not have a unique solution for the current, as any zero-resistance loop in the circuit (such as two super-conducting wires running in parallel between the same two points) can support an arbitrary circulating current. --catslash (talk) 02:02, 31 October 2013 (UTC)[reply]

For a classical linear electrical network (ie with idealized resistors, capacitors and inductors) the problem of determining the currents through the elements given all the applied emfs always has a solutions and the solution is unique. This has been known/proven for decades (starting I think from Gustav Kirchhoff's work) and is not a empirical result. You should be able to find the proof in any univ. level electrical network analysis book; for example look up Mac Van Valkenburg's text. Since I don't have access to it right now, here is a more mathematically sophisticated proof of the result. Abecedare (talk) 02:27, 31 October 2013 (UTC)[reply]

The above-linked text Mathematical Aspects of Electrical Network Analysis does not claim that every classical linear electrical network has a unique solution for the currents, only those networks which are what it calls ohmic. Also, the definition it gives for ohmicity seems to be little different from being soluble for the currents - at any rate, any definition in terms of the circuit topology or component values is well hidden. --catslash (talk) 18:56, 31 October 2013 (UTC)[reply]

You are right,I should have been clearer.

The result I stated for an idealized RLC network and (independent) voltage sources is true but needs one topology condition to eliminate the degenerate case of connecting say two identical voltage sources to each other and asking what current is flowing through that loop (for idealized sources the current is indeterminate). So the result holds as long the circuit doesn't have any loops with only voltage sources.
The result also holds when we introduce independent current sources into our circuit, as long as there is no cut set containing only current sources.
The result can be further extended for circuits with mutual inductances (transformers), with the additional conditional that no set of these inductances be "perfectly coupled".
Roth considers an even more general case with dependent sources and any general linear elements and formulated the condition of "ohmicity" (which can be shown to reduce the more intuitive forms for the special cases listed above).

All of this is covered (with proofs) in modern electrical network analysis textbook, and the simplest case asked by the OP will be found even in older "classical" texts like the ones Valkenburg, Seshu & Reed, Chen, Balabanian & Bickart etc. What I want to emphasize, is that none of this is taken on faith or considered magical like magnets. :) Abecedare (talk) 00:01, 1 November 2013 (UTC)[reply]

Your conditions dispose of the overdetermined cases of incompatible sources, and so guarantee the existence of a solution. What about the underdetermined cases? --catslash (talk) 17:25, 2 November 2013 (UTC)[reply]

The listed conditions in fact guarantee a unique solution.

Further: for the "Roth" case the ohmicity condition is just sufficient for the existence and uniqueness of the result (it becomes necessary too with some additional assumptions). For the rest, the given conditions are both necessary and sufficient. I found an online source (see pages 125-132) which discusses/derives results of the sort listed above; note that for pedagogical reasons, it starts with the general ohmicity condition and then reduces to simpler conditions for the special cases. IIRC some classical texts take the opposite approach: state and prove the RLC result and then relax conditions for more general cases. The latter may be easier for the OP to follow, but unfortunately I didn't find any of them accessible online. Abecedare (talk) 20:18, 2 November 2013 (UTC)[reply]

Sorry, I misread your third condition (imperfect coupling) as being necessary only when coupled inductors are present.

Wai-Kai Chen's Applied Graph Theory which you link to seems to define perfectly coupled as: having a zero eigen value. If I look for and find a zero eigen value, then I have already found a set of non-unique solutions, namely any multiple of the corresponding eigen vector (possibly plus some other stuff, if there are sources). So the uniqueness part of the theorem reduces to: the solution is unique unless you can find a non-unique set of solutions.

Whether or not there is a simple test for ohmicity/perfect-coupling/uniqueness that does not amount to solving the system (unlikely), it is nevertheless easy to construct circuits that have non-unique solutions (at any given set of discrete frequencies (which could include d.c.)), so in general a circuit will not have a unique solution.

It should however be possible to put a bound on the number of frequencies at which non-unique solutions might exist, based on the number of capacitors and number of inductors in the network. --catslash (talk) 02:02, 3 November 2013 (UTC)[reply]

"the solution is unique unless you can find a non-unique set of solutions": That is of course tautologically true but not a useful or suggested way to analyze the problem; and the above tests do not do so. And developing a "simple test for ohmicity/perfect-coupling/uniqueness that does not amount to solving the system" far from being unlikely, is the whole raison d'etre of the above discussion and the book section! For example, for an RLCM case, note that:

The matrix L whose non-singularity needs to be tested is only a sub-matrix of the bigger matrix Z_pp, which itself is only a sub-matrix of the big "system matrix" in equation 2.271 that incorporates all the information about the specific circuit along with the KCL and KVL constraints.
Checking for non-singularity (or bounding a positive-semidefinite matrix's smallest eigenvalue value away from zero), is much easier computationally than computing its eigen-decomposition explicitly.
As the book suggests in the last paragraph on page 131, even if the naive impedance matrix turns out to be singular, there are workarounds for obtaining unique solutions for the current through branches of practical interest.
Even in instances when the solution is non-unique (eg, is say a transmission line or microwave circuit with multiple frequency modes), the results listed above are useful to characterize the solution space, and engineering techniques have been developed to damp the undesirable modes of operations.

Obviously the textbook doesn't go into details of the last two points, but such methods are old-hat by now and used routinely in electronic circuit and electrical network simulation, design, and operation. Frankly, the basic existence/uniqueness problem is so well-solved that this is somewhat of a boring area for fresh research, because one needs to invent really exotic scenarios to break existing methods. That's the reason this 2012 book is not citing any work later than 1967 in the existence and uniqueness section! Abecedare (talk) 06:29, 3 November 2013 (UTC)[reply]

You are entirely right and I was wrong about the ease of calculation of perfect-coupledness. However the proof seems to require that the circuit be initially relaxed (p127), and is actually merely demonstrating that the solution is unique if you ignore voltages and currents that (for a real frequency) are not delivering power from the sources into the resistors - a result which is entirely expected. A circuit with no sources might support any number of degenerate modes, but under this criterion of admissibility only the trivial (zero) solution would be recognized. Unfortunately, the viewable section of the book is not self-contained, so it's possible that I'm still misunderstanding something. --catslash (talk) 13:09, 5 November 2013 (UTC)[reply]