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Posts tagged with dynamical systems

supplysideliberal:

[M]y only qualifications … are having read several books about marriage and experience from my own marriage, now in its 30th year…
  • “… a misconception: … to make a marriage work, you have to find the right person. The fact is, you have to be the right person,…
  • … students … interview friends about their … weaknesses, and discuss what triggers their … reactions … in order to understand their … hot buttons… “Being blind to these causes people to experience problems as due to someone else—not to themselves,” Solomon explains. “We all have triggers, blind spots, growing edges, vulnerabilities. The best thing we can do is be aware of them, take responsibility for them, and learn how to work with them effectively.”

  • … blaming, oversimplifying, and seeing themselves as victims are all common traits of unhappy couples and failed marriages. …
  • frame statements as “X, Y, Z” statements, rather than finger pointing: When you did X, in situation Y, I felt Z….

Here are a few of my own thoughts on marriage: 

  1. There are a huge number of dimensions on which one might wish to be well-matched with one’s spouse. There is no way you are going to be well-matched on all of those dimensions.
  2. The reputation you have built up with your partner for telling the truth about objective facts is a precious asset in any relationship. …[T]here is bound to be some way to tell the truth. (If you can’t think straight, say “I can’t talk about this right now,” rather than lying.) The more subjective realm of revealing what is in your heart is trickier; seize moments when you will be able to express yourself well and be well understood. It is worth working toward being known.
  3. In an argument, if each partner comes back with 101% of the irate heat the other just gave, things will explode. But if each partner ratchets down the intensity to 99% of the intensity of the last remark, things will eventually calm down.* So a small difference in reaction pattern can be the difference between an explosion and something that simmers down.

    

* Math note: To pursue the logic a bit more, if your partner is coming back with 125% intensity on each round, you are going to have to return less than 80% intensity on each round to avoid an explosive chain reaction….

A regular reader writes in:

I tried that for a long time. It was better than escalating in the short term. But I eventually realized that some people like explosions, and created them when they weren’t happening.

…having been married for a long time, I can guarantee you, that no one on the outside has any idea what goes on inside a marriage.

The parts of being married I enjoyed were great. I’ll never do it again.













I had judged The Emperor’s New Mind by the negative reviews but never actually picked it up. It has a lot of great stuff, almost like an “early draft” of The Road to Reality.




All I knew about Emperor’s New Mind before was that it invokes quantum mechanics to explain free will, which was perceived as “icky” by people who study the brain. (Despite that, like quantum nonsense, the “greats” of QM—Bohr, Schrödinger—also weighed in with QM/free-will speculations (do you hear me, Conrad&Kochen? Quantum communication folks?) — because, let’s be real here, free will is a millennia-old conundrum and I think we’d all appreciate it if the people who understand compositions of Hilbert spaces weighed in on whether and what the latest “master theory” (bringer of semiconductors = transistors, LCD’s, lasers, MRI/PET and certain polymers/piezoelectrics/other materials) would say about the age-old question)

I got a bit more of the debate whilst reading about pi-1 sentences, which is a computability/knowability/logic dealio. But again, this was the level of “What’s RP’s argument in a nutshell?” rather than “Is here anything worth reading in the 400 pages?”. It’s a lot of good.










@IgorCarron blogs recent applications of compressive sensing and matrix factorisation every week.
(Compressive sensing solves underdetermined systems of equations, for example trying to fill in missing data, by L₁-norm minimisation.)
This week: reverse-engineering biochemical pathways and complex systems analysis.

@IgorCarron blogs recent applications of compressive sensing and matrix factorisation every week.

(Compressive sensing solves underdetermined systems of equations, for example trying to fill in missing data, by L₁-norm minimisation.)

This week: reverse-engineering biochemical pathways and complex systems analysis.




The word “Evolution nowadays suggests "evolution of organisms by natural selection as per Darwin & modern population genetics".

But what about other kinds of evolution? Any unary endomorphism from a system onto itself, applied over and over to generate “time”, could be considered a kind of “evolution”.

  • Crystals and quasicrystals evolve naturally.
  • Caves and stalagmites evolve naturally.
  • History evolves artificially … although no-one knows the mapping.
  • Art evolves artificially … again, no-one knows the mapping. (but we know there is cross-pollination. Could we call it “Art sex”?)
  • Proto-biological chemical compounds, like basic amino acids, “evolved” by a method similar to natural selection.
  • Businesses (and entrepreneurs) that grow through trial-and-error, evolve their ideas and their business processes by artificial selection.
  • Romantic relationships evolve. Any human relationship evolves. Sometimes “a” relationship can evolve with a group of people as-a-unit. But here I’ve found the selection to be more exogenous than endogenous. Friendships seem to pick up exactly where they left off γᵢ(Tᵢ) rather than be pitched to the dustbin of 0. Even romantic relationships seem to hang mostly where they were, even after a breakup. Perhaps the breakup zeroes the romantic part ⌊γᵢ⌋ᵣ, but everything else—sexual chemistry, personality dynamics, humour dynamics, and history—remains stubbornly unbudged by a breakup per se.
  • And like I said, any endomorphism, repeatedly applied to a system, could count as “evolution”. (An endomorphism draws both the input and the output from the same domain, i.e. ƒ: X→X.) If there is a throw-away criterion (mapping ↦ 0), we could call that “selection”. Any fixed point of the mapping ƒ(p)↦p is an endpoint of evolution.
     

In this video, John Baez talks about how the inherent interestingness of the number 5 has made itself apparent to us through several processes:

  • artists (mosque designers) discovered it. God speaks to us in the language of mathematics, remember?
  • crystals and quasicrystals discovered 5 by evolution — but not the biological kind
  • soot and space dust found , also by natural non-bio evolution
  • BTW, unrelated but some Scots (perhaps Picts) carved some Platonic solids out of stone centuries before Plato … so perhaps they should be called Scottish solids.
  • the Pariacoto virus found by biological evolution
  • Roger Penrose (a mathematical physicist) discovered & described 5-way symmetry in modern mathematical (group theoretic) terms

In each case, logic is the canvas. Art — nature — mathematicians are the painters.




  • Are the Israelis bellicose because countries are always antagonizing them, or are countries always antagonizing the Israelis because they’re bellicose?
  • Do you think that boy is cute because he likes you, and he likes you more because you like him, and where did this liking start in the first place?
  • If the rich get richer, how do you get rich in the first place? And if poverty leads to violence, lack of education, and ill health, which leads to more poverty, how do you stop the vicious cycle?
  • Do cops mess with people because there are too many criminals around, or do people become uncooperative and untrusting of cops because they’re always messing with people?
  • Nature or nurture?
  • Do Americans commute long distances to work because the country is built around roads, highways, and the assumption that everyone has a car, or is the highway infrastructure good because Americans have so many cars?
  • If child abuse is caused by the parent being abused as a child — if gang violence is usually retaliation for other gang violence — where did it all begin?
  • Does the culture generate the media, or does the media generate the culture?

So many chicken-and-egg questions. Dynamical systems theory comes to the rescue.


the Lorentz attractor

A dynamical system is a time-evolving list of equations that mutually affect each other. In the real world the arrow of causation often runs both ways. By thinking about chicken-and-egg questions as a dynamical system, suddenly circular causation becomes logical, even quantifiable.

fig3

More technicals on linear operators on dynamical systems, some famous examples including ecosystem modeling, mind-body dynamics, Jon Gottman, toy models, and the similarity to (I thought boring) differential equations later.




 
ƒ(ƒ(ƒ(ƒ(x))))
 
Above pictures the phase-space of some observable, x, that evolves over time x₀, x₁, x₂, x₃, ….  The next time step after value x₃ has been reached is shown as the height of the graph above x₃.
(So unlike the graph of a baseball thrown from the origin to (x,y) = (1,0), this is not a time-path picture.)
Also notice that, at whatever time the process might reach, say, x₇ or x₅₅ = .34197, the same thing will always result next turn:   x₈ or x₅₆ will become ~.6.
So time in this picture proceeds by jumping to parabola, diagonal, parabola, diagonal, parabola, diagonal, parabola, diagonal.  When one reads the width of the parabola, or the height-or-width of the diagonal, one knows the value of the process at this particular time step.
What’s happening mathematically is that the transformation 

is being applied over and over again.  So ƒ(ƒ(ƒ(.5))) represents three time steps after starting in the middle.  ƒ(ƒ(ƒ(ƒ(ƒ(ƒ(ƒ(0.000001))))))) represents seven time steps after starting at the left side.  This is a spiffy way to describe how systems behave, even if it’s just a one-dimensional observable.  Iterative mappings, or “difference equations”, are a popular way to describe real-life systems.  I first saw it in economics class, in Robert Solow’s model of capital growth.  It also appears in weather models, psychological models, and biological models.
Question.  What happens if you do f(f(f(f(f(f( something ))))))   ad infinitum?  Do the answers ever settle down?  This is where, if you had a button on your calculator to do “apply f”, you would hit the button and see what happens.  Unless you can program, though, you’ll have to think through this one logically.
x̄̄ is a stable fixed point of this iterative mapping.

However consider a very similar cousin,
.
In the .9 case, x̄  is unstable.  What’s up with that?

ƒ(ƒ(ƒ(ƒ(x))))

Above pictures the phase-space of some observable, x, that evolves over time x₀, x₁, x₂, x₃, .  The next time step after value x₃ has been reached is shown as the height of the graph above x₃.

(So unlike the graph of a baseball thrown from the origin to (x,y) (1,0), this is not a time-path picture.)

Also notice that, at whatever time the process might reach, say, x₇ or x₅₅ .34197, the same thing will always result next turn:   x₈ or x₅₆ will become ~.6.

So time in this picture proceeds by jumping to parabola, diagonal, parabola, diagonal, parabola, diagonal, parabola, diagonal.  When one reads the width of the parabola, or the height-or-width of the diagonal, one knows the value of the process at this particular time step.

What’s happening mathematically is that the transformation 

f(x) = 2.8 x (1 - x)

is being applied over and over again.  So ƒ(ƒ(ƒ(.5))) represents three time steps after starting in the middle.  ƒ(ƒ(ƒ(ƒ(ƒ(ƒ(ƒ(0.000001))))))) represents seven time steps after starting at the left side.  This is a spiffy way to describe how systems behave, even if it’s just a one-dimensional observable.  Iterative mappings, or “difference equations”, are a popular way to describe real-life systems.  I first saw it in economics class, in Robert Solow’s model of capital growth.  It also appears in weather models, psychological models, and biological models.

Question.  What happens if you do f(f(f(f(f(f( something ))))))   ad infinitum?  Do the answers ever settle down?  This is where, if you had a button on your calculator to do “apply f”, you would hit the button and see what happens.  Unless you can program, though, you’ll have to think through this one logically.

̄ is a stable fixed point of this iterative mapping.

x_{k+1}   =   x_k  (1 - x_k)  4mu

However consider a very similar cousin,

x_{k+1}   =   x_k  (1 - x_k)  4mu.

In the .9 case,   is unstable.  What’s up with that?


hi-res