{"id":795,"date":"2009-07-13T17:55:00","date_gmt":"2009-07-13T17:55:00","guid":{"rendered":"http:\/\/jimgworld.com\/blog1\/?p=795"},"modified":"2010-05-14T21:25:01","modified_gmt":"2010-05-15T02:25:01","slug":"795","status":"publish","type":"post","link":"https:\/\/jimgworld.com\/blog1\/?p=795","title":{"rendered":"STRANGE MAGIC"},"content":{"rendered":"

I\u2019ve been interested in chaos theory for almost 20 years, ever since reading James Gleick\u2019s \u201cChaos, Making of a New Science\u201d in 1990. I\u2019m not a mathematician, but I was able to appreciate most of the general concepts that scientists were finally beginning to consider when chaos started to become popular. E.g., the \u201cbutterfly effect\u201d, how a butterfly flapping its wings in a certain way over Beijing can influence the track of a hurricane over Jamaica six months later. <\/p>\n

Today the \u201cbutterfly effect\u201d has pretty much become folk wisdom, although the folk view often goes too far. (E.g., if only that darn butterfly went left instead of right, New Orleans might have been spared!). Chaos study shows that many physical systems are non-linear in nature and are recursive in nature. They are thus best described by math equations that are themselves non-linear and recursive (i.e., they utilize the recent outputs from the system as one of their input streams; the past never completely goes away for them). Under certain conditions (and NOT generally), these equations can be very sensitive to initial input assumptions. Again, in certain limited situations (such as the weather), a tiny change in an input factor, say about a tenth of a percent worth, can send the output up or down by maybe ten percent. <\/p>\n

People who have reason to be cynical about science, e.g. New Age types and spiritualists, often cite the \u201cbutterfly effect\u201d to discount all of science and give their own \u201cmystic intuition\u201d more credence. Well, they have their agendas. Such people often favor holistic remedies, but if they get a really tough infection that threatens their life, they usually take the antibiotic pill that scientists found to be 99.9% effective. They trust that the butterfly was not flapping in this instance.<\/p>\n

What I\u2019ve always found more interesting about chaos theory, and harder to understand, is the notion of \u201cstrange attractors\u201d. (Here\u2019s a typical example<\/a> of a strange attractor explanation; if you get it, you\u2019re smarter than I am \u2013 but don\u2019t crow, because I\u2019m not all THAT smart). But over this past weekend, while reading a book about chaos and brain dynamics (\u201cHow Brains Make Up Their Minds\u201d by Walter J. Freeman), I finally managed to wrap my mind around it! Eureka! I\u2019m only twenty years too late.<\/p>\n

The best way, I found, to understand strange attractors is to compare them to with the two other main types of attractors<\/a>. The first kind, the simplest, is called the \u201cpoint attractor\u201d. When a system has a point attractor, the system is always trying to get back to one particular fixed state. The classic example is a pendulum at rest. We\u2019re talking about pendulums as they really exist here on earth, with friction in the joints \u2013 that\u2019s an important point here. The earthly pendulum is at rest when it is straight up and down. You can flick it and it will start swinging back and forth. But because of friction in the joint (and from the air), it will soon come back to rest, straight up and down. That\u2019s the \u201cattractor state\u201d of it, a state described by a single point (i.e., the point where the pendulum bob is closest to the ground, with its arm straight up). <\/p>\n

The second kind of \u201cattractor\u201d is a limit-cycle. That\u2019s when the system \u201coscillates\u201d, i.e. moves in a circular pattern of some sort. This circular pattern is perfectly predictable. If we could somehow eliminate all friction in the pendulum joint and remove the air around it, you could flick the pendulum and it would swing back and forth forever. If you plotted the pendulum\u2019s position along its arc versus its speed, you would get a perfect circle. For whatever reason, the science and math people decided to call this a \u201climit cycle\u201d. <\/p>\n

In the real world, an old-fashioned cuckoo clock or grandfathers clock is a system having a limit cycle. You pull the cord to lift up the weights, and the pull of gravity on those weights is transmitted by some gears and springs as to give the pendulum a little kick, just enough to overcome friction in the air and in the hinge. So the clock\u2019s pendulum goes back and forth predictably, and thus powers the internal gears that drive the clock hands. If all the gears and stuff are designed well (by German engineers and craftsmen, no doubt), and you keep pulling up the weights now and then, the clock will keep good time. The limit cycle will be maintained.<\/p>\n

(Yes, I\u2019m old enough to remember cuckoo clocks. My grandparents had a big one in their apartment, and my parents had a smaller one in our kitchen. When you woke up in the middle of the night, you were never more than 59 minutes or so from finding out what time it was. But the constant tick-tick-ticking of the pendulum often helped you get back to sleep before the mechanical cuckoo sang its hourly song.)<\/p>\n

So what is a \u201cchaotic attractor\u201d, more colorfully known as a \u201cstrange attractor\u201d? Well, that\u2019s the next step up in the sequence from point attractor to limit-cycle attractor. A strange attractor mixes chaotic unpredictability with just enough regularity to still be considered an \u201cattractor\u201d. To make a cuckoo clock \u201cstrange\u201d, you would have to get a mad clockbuilder to put in some additional gears powered independently by a separate weight, and have the output of these gears join in somehow with the regular \u201ckick\u201d that goes to the pendulum. These \u201cstrange gears\u201d would work out of synch with the original set of gears; the pendulum might get a kick one way according to one frequency, then get a lesser or greater kick in the other direction at a different frequency. <\/p>\n

As a result, the pendulum (and thus the clock) would be all mixed up. Sometimes it might be a half second between ticks, at other times it might take two and a third seconds. So now there is chaos. But it\u2019s not absolute chaos; the pendulum is still ticking, after all. It doesn\u2019t stall and stop ticking, nor does it keep speeding up until it breaks (although that could be arranged; but we are assuming that our mad clock scientist didn\u2019t put that much ooomp in the \u201cstrange\u201d kicker system). <\/p>\n

Perhaps this clock still keeps some kind of time, but not accurate time; it\u2019s almost always ahead or behind the real time, and you\u2019re never sure which. But on average, over many minutes and hours, the clock hands still advance at a steady rate; this average could even be 24 clock hours per day. So the demented \u201cstrange cuckoo\u201d could still be \u201cattracted\u201d to a 24 hour day, but would be too chaotic within that day to give you accurate time. Its \u201climit cycle\u201d has gone over to become \u201cstrange\u201d. But it still has a beating pendulum that hasn\u2019t flown off the hook (that would be a further \u201cstate transition\u201d for this crazy cuckoo clock). <\/p>\n

So that\u2019s the strange magic of strange attractors. I\u2019ve been fooling with an Excel program that develops strange attractor patterns for an interacting math system that describes something like a clock pendulum; I\u2019ve attached some of the \u201cstate space plots\u201d below. Each reflects the output results over time from a particular set of input conditions. This is just a small sample of the many possible \u201cstate space\u201d patterns that can occur between position and velocity, as the input parameters are \u201ctweeked\u201d. Sometimes a small tweek has hardly any effect on the pattern. Once in a blue moon, a small tweek changes everything \u2013 and I see that chaotic butterfly at work! It\u2019s all quite interesting \u2013 and quite strange!<\/p>\n

<\/div>\n","protected":false},"excerpt":{"rendered":"

I\u2019ve been interested in chaos theory for almost 20 years, ever since reading James Gleick\u2019s \u201cChaos, Making of a New Science\u201d in 1990. I\u2019m not a mathematician, but I was able to appreciate most of the general concepts that scientists were finally beginning to consider when chaos started to become popular. E.g., the \u201cbutterfly effect\u201d, […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[],"_links":{"self":[{"href":"https:\/\/jimgworld.com\/blog1\/index.php?rest_route=\/wp\/v2\/posts\/795"}],"collection":[{"href":"https:\/\/jimgworld.com\/blog1\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/jimgworld.com\/blog1\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/jimgworld.com\/blog1\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/jimgworld.com\/blog1\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=795"}],"version-history":[{"count":4,"href":"https:\/\/jimgworld.com\/blog1\/index.php?rest_route=\/wp\/v2\/posts\/795\/revisions"}],"predecessor-version":[{"id":1530,"href":"https:\/\/jimgworld.com\/blog1\/index.php?rest_route=\/wp\/v2\/posts\/795\/revisions\/1530"}],"wp:attachment":[{"href":"https:\/\/jimgworld.com\/blog1\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=795"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/jimgworld.com\/blog1\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=795"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/jimgworld.com\/blog1\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=795"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}