From enga Wed Aug 23 11:19:03 1989 I'm not an expert in your field. My naive observations are based on my extensive experience of failures in dealing with other sciences. seems that your group is well divided into two parts. There can be no (?) line between the mathematicians and the applied people. And, of course, this is (?) to physics and the other sciences. It's kind of a universal structure. And my experience in general is that the connections of these two groups in a loop provides a cycle which is the fundamental growth cycle of the history of the sciences. So, when these connections are weak, then growth is slow. Now in your particular case, this is speculation-not totally justified-but say this is just a fantasy for the sake of fun, that the applied group has particularly to do with chaos. While the mathematical group is particularly addicted to oscillators. So this is a kind of generalization of the traditional dichotomy between order and chaos that underlies the Judeo-Christian system. Therefore, the attempt to re-map these things, it is appropriate that it should be introduced to this group by (?). And, indeed, in the physical sciences we see that Start editing: The advent of chaos theory has brought about a new reconciliation between mathematical problems and the applied physical sciences I'm going to organize my talk in two parts, one for each of these groups: first, on forced oscillators for the sake of the descention of the theoretical growth in the direction of data; and, secondly, on the outreach of the applied group toward more theory. I will try to give just three short samples of theory on each of these parts. 1. The zoo [?]of oscillators and some ideas on the impressions of oscillators and the occurence of chaotic behavior in forced oscillators. 2. Under chaostrophe: the ideas of fractal dimension for chaotic data about the reconstruction of the alpha (?) models for chaotic data. 3. About the possibilities for forecasting the alpha (?) data (or is it alpha beta?). If I succeed in covering this long program in about 30 minutes it would be a miracle. So, I'll begin with a little history of the all and everything. We'll start with Helmholtz. Helmholtz was a great and noble emperor of Europe [where?] in the last century. One of his minor accomplishments was the invention of the doorbuzzer. His design is still used today. It involved a tuning fork and a piece of wire wrapped around it in a bowery (? or a battery?), and when you close the circuit it buzzes. Shortly afterwards Lord Raleigh, who was very fond of making mathematical models, tried to make a model for musical instruments. He made models for the clarinet, the flute, the organ pipe, and so on. One of his greatest models was for the violin bow. Without a doubt, he was the Goodwin of mechanical engineering. He also made a model for this doorbuzzer which is the so-called Van der Pol equation, invented by Lord Raleigh to model this buzzer. And that was the start of the subject called nowadays non-linear dynamics which at that time was known as the theory of oscillation. Then, we have Paul Gray, which Thala(?) has told us about his winning the Oscar. He studied the solar system to try to understand it Will the sun rise tomorrow? That's the question. If you prove in a mathematical model that the sun would rise tomorrow, we could then in fact have faith that it actually was gonna rise. Science has made fantastic advances in theory in order to tackle this difficult problem. Some minor techniques he invented included a whole differential topology that involved an extension from the two dimensions of Raleigh's model for the doorbuzzer, because three dimensions was the minimum necessary to incorporate the properties of the solar system. Most of the examples I'll describe today are three-dimensional and his techniques are important. Then, experimentalism arrives (???) and you think the --- pendulum, big pendulum hangs the little pendulum and this was an experimental laboratory for the theory of oscillations, which dominated the subject up until very recently with the advent of analog, ??? analog computers. Somebody invented the radio transmitter, I don't know if it was Mark --- a glass tube with three metal parts inside, and somebody figured out how they could oscillate. Then it was necessary to learn how to put information on the oscillation and how to extract it. Van der Pol, in Holland, was the person fundamentally associated with the invention of the FM detector, which makes radio transmission from Moscow to Copenhagen and so on possible. He was also an experimentalist. He studied the mathematical model of the Raleigh oscillator forced by an external oscillator which (could hear?) your voice, and he experimented in theory. ??? experiments for which they didn't have telescopes in those days, so the observation of the ??? oscillation was done with earphones, when it was oscillating you'd hear a tone. One day he picked this up,and after tuning for a certain condition he heard a buzzing in the earphone. He knew he had recorded and carefully drawn the experimental data, and so he actually observed chaos in the (?) Van der Pol system. He might be the only one who ever has, because this particular system resists chaos. So now let's skip to 1960s or so. We ought to look at 1940 first - I should bring Birkoff in first...But I'm going to leave Birkoff out because, while he made the first mathematical study of chaos after Poincare, it was in the context of the solar system. Here, I want to stick to the history of forced oscillators just to keep it finite. Hiyachi perfected the technique of using analog computers to explore non-linear dynamical systems, and he made great discoveries. One of them I think was his student Ledda's (Ueda?) first observation of chaos in a carefully measured experiment with analog computers. Many experiments extending from that time to the present. He preceded Lorenz of the Lorenz equation, and then in 1970 Ruelle Takens completed the application of chaos theory to physical systems and started the fantastic chaos revolution going on today. Now I'll give you some examples of oscillators so you won't think that the Van der Pol oscillator is the only one. First of all, in the pendulum clock or the pocket watch ??? mechanism we have a nice mathematical model due to ???Domhoff?? one of the greatest ??? systems theorist in the Soviet Union in 1930. The harmonic oscillator is a conservative system, it has energy levels that would be preserved by a harmonic oscillator, that is to say a pendulum without friction. In case of friction, the energy is decreased. In a harmonic oscillator with friction the energy gradually runs down to zero. Galileo discovered the so-called isochronous principle of the pendulum in church while the other people were praying. He was watching the lamps on the pew and noticed that no matter how much the amplitude grew and shrank with each successive turn, it always took about the same time for the oscillation, and that's the principle on which the clock is based. Therefore, after running down a long time you could give it a kick of energy, putting the energy in and raising it back up to a higher oscillation. This is the trajectory of a limit cycle of the ??? pendulum clock. Completing the equations and everything (???) so that's an important oscillator to know about. THE FOLLOWING DESCRIBES DIAGRAMS, CAN'T EDIT IT. As we'll see in a few minutes, the fact that the shape of this has a hook on one end is very important. It's not that every oscillator has a limit cycle and it doesn't matter if the limit cycle is round or square.The actual shape is sort of geometro-dynamics. The shape matters in applications and the geometry of oscillators is important. This clock wouldn't keep time if it weren't for that hook. Another interesting one is a recent discovery - the Kadyrov oscillator. It consists of two cusp catastrophes. You know what this is, I think it's a response diagram for a dynamical system with one safe variable that goes this way and two control parameters and horizontal planes and well let me go down here to the control plane and in ??? planes there is a cusp shaped curve so that when inside- controls are inside of this region there are two possible ???? and ??? there is only one. Where in here, this is the locus of the ???(fracture?). Here, this surface is repellent. So this was presented in the 1970s by Zeeman as a model for many different social processes and in one particular case a model for hawks and doves emerging in a democracy ??? government in a state of war or less aggressive posture toward neighbors. So, Kadyrov had ??? two of these models for an arms race model involving two nations. Let's follow - connect the state of one to the back and forth control parameter of the other and vice versa. Now if you did this in such a way that there end up four couples, fully coupled systems, four different parameters. One, so this has two parameters and this does and after (or actually?) one of these parameters is controlled by the other system then one parameter is no longer ??? (programmed for advantages?). Likewise this one. ??? two. The increase, he introduced two new parameters for the (strength?) of these countries. And (in fixing?) two parameters at reasonable values and examining the behaviour of the coupled system under the action of the other two he found this behavior. First of all, this fully coupled system, in case A=B, so (it's a symmetric?) case, is still in the category and in this context of category theory and is known under a variety called the parabolic umbilic. So, on the grounds of this theory it's known that it should be two cusps - something like this is one and this is the other you get an amplitude (or aptitude) of resulting diagram. ??? in my system (?) it might be four (Saturday crackers/tractors?), out here two, out here two, out here one, out here one. One, one, one, two, two, and there four tractors?. Well, when A is different than B algebraic geometry no longer applies because we're outside the context of category theory a few (or new?) modifications are introduced. One (he found ?) is here is sort of a parabolic shaped curve and here likewise is another one. And in this region is an oscillation. So in this case, according to the model anyway ??? in which the armaments or the amount of GNP devoted to armaments would be oscillating as the years go by between the two nations. Here's another novel case. William Smith of Canada attempted to model the onset of puberty in mammals. He works at a university that has a very large and famous veterinary department. The people there use bulls and rats and rabbits to take blood samples and analyze them, and they came up with a physiological model for the endocrine system that controls reproduction. There are three elements: the hypothalamus, the pituitary, and the gonads. The hypothalamus communicates to the pituitary by sending a chemical out called lataraider? routinizing the hormones, release hormones. The pituitary communicates to the gonads by sending another chemical messenger called routinizing? lutinizing? hormone. The gonads communicate back to the hypothalamus by sending either estrodyal in case of females, or testosterone in case of males. The mathematical model that Smith created for this situation involved a complex system with an identical response diagram for each system that's simply a lader? attractor so exponentially ??? to people ???? for homeostasis with control monitor adjusting the position of that homeostasis at higher or lower levels of release. The coupling of these systems expresses the control parameter tied (?) to the homeostasis of this one as a function of the internal state of ??? chemical of this one through essentially a linear function but with a floor and ceiling. So you have a graph of this coupling motion that looks like this: ????. Likewise here, this one is negative because the testosterone inhibits the release of (stimulants to the cells?). And in this model Smith put a parameter for the steepness of this coupling motion - it's the only parameter and the result is a (hopladder cation?) ??? the state space which is three dimensional we'll call these cariables X, Y and Z. And then we have here a three dimensional space variables X,Y and Z but changing the changing view gives a face portrait that (depends?) this parameter and for smaller view, less steepness of sensitivity of the hypothalamus to feedback, you have homeostasis attracted ??? but after passing a certain critical value it begins to oscillate. This explains puberty which is understood by physiologists as a oscillation in the levels of these three hormones in the blood with a period of about 6 hours for humans. So before puberty there's homeostasis with these chemical transmittors and after puberty there's oscillation with a period of 6 hours. So this model is a familiar one from the (zoo of bifurcations called Hop bifurcations?). So Smith went barnstorming around the world giving lectures called puberty ??? bifurcation. .LP It's known physiologically that this sensitivity increases with age. So the model actually fits the data ?? course ?? chaotic and the model is ???. So then working with Smith later on I added (another/an order of?) feedback going this way which are the physilogical spaces in the transmission of this chemical from the hypothalamus to the pituitary. They're very close by and the communication is done with a very rapid blood flow through a small channel which is turbulent. So there is a backflow. And adding this made this chaotic but furthermore introduced another state so that there's more complicated bifurcation diagram and shortly after puberty it's possible to end up with either a smaller oscillation or a larger one which is then the model for amenoria? a pathological state in which puberty exists but the oscillations are too small and suggesting therefore an intervention therapy for this disorder called amenoria? by giving a shot of one or another of these chemicals ??? that would bump one from the base of attraction of the pathological oscillation to the base of attraction of the normal oscillation. This was tried out in a medical clinic in San Francisco and found to work. So, this talk would be better if I could show you a computer demonstration of the actual oscillators so you could see that their shape is very, very unlike a circle and has more bumps than this. Things are like a smoke ring. It's very stretched out, and turned in spaces of higher dimension. .RESUME EDITING Next, I want to talk about the oscillators when one adds forcing. First we'll consider weak-forcing. The results of ?? are called incrama? Later we'll look at strong forcing. In case this one is forced we found that (observed by ???) we have two models. In fact he [who?Van der Pol?] had a whole bunch of them because he had a clock maker in Amsterdam working for him full time trying to develop the perfect clock so as to extend the navigational capability of the Dutch ??? navy, so that the Dutch empire could be extended to the maximum. There was international competition for the best clock for the chronologer. He had a lot of different examples hanging on the wall, on the table and so on. He observed the two pendulum clocks on the table in the living room and no matter what he did, the pendulums were always swinging in the same phase. So even though neither of these two clocks was keeping good time, each of them was keeping the same time because not only did each pendulum complete one cycle in the same period of time but always in the same phase. He used to kick one to see how long it would take to return to the same phase. It took a few minutes. He recorded the observations. He found that in some cases he could get the synchrony? in opposite phases, but most of the time in the same phases. So that's called phase entrainment. I want to explain that in terms of the geometry of this picture, and this explanation is recent. There's a theorem of ??? around 1960 that says this should happen, at least you should get a frequency in entrainment. I'll draw a picture of it in a minute. But, ??? theorem's didn't explain phases. According to ???'s theorem you could have this or one pendulum out there at the end of the cycle and the other's in the middle and then - you know what I mean? (At fixed delta fee?) phase difference would be maintained indefinitely. But why should delta fee be zero or under 3 degrees? So here's an explanation of Mozelo Ferrara around 1980. We have the two clocks on the table ??? one clock and the other clock has a ? mechanism which goes tick tock. And when it goes tick tock it sends a solitary wave through the table and gives a perturbation in the only possible direction to the other one. And this perturbation can then be intermitative? as a pulse ??? pulse tidal force apply to this picture which goes this way. Because this is displacement direction and this is velocity direction and the vector direction of the velocity is ??? accelerational force. So (while you're looking/moving?) down here on a trajectory and suddenly comes a pulse in that direction you're moved back on the ?. And when you're coming up this way, suddenly comes a pulse in that direction you move up. Either way it moves up. So the phase gradually moves around cycle to the top. And if it had happened that this clock had a in its original phase portrait two hooks which could be a possiblity then there would be two possible favorite phases under a periodic pulse (tile ???) in that direction you could get stuck in this phase or stuck in this one. So the natural shape of the limit cycle has enormous implications for behavior, periodic behavior, in a context of forced oscillation. ?? forced oscillators ???. Furthermore, this kind of picture where models are taken from the scientific literature for different subsystems and then connected together in this connection as paradigm in which the main parameter again under the control of the doctor, the ministers, is some parameter associated with the strength of (coupling?) the connection as paradigm. Here if you just had forces ?? you could imagine these two cooperating in oscillation. Well these two cooperate in another oscillation. And then, as the strength of the coupling between the two oscillators is increased you get first of all this phase entrainment which, depending on the direction of the periodic pulse, could result in one or a different ? phase of ??. . Now, I want to force this picture, without- this is already forced by the escapement? mechanism to maintain the oscillation, but let's - in order to describe what you do ???. Consider this phase portrait of the completely dying-away pendulum and force it with an ecloginist? oscillation by whatever means. The good picture in this case invented by Poincare is this ??? ring model so we have the original variables X and X' in this picture, but now we have the phase of the forcing oscillations as a third variable. It's a periodic variable. So we take this space of X and X' and swing it out and around a circle and bring it back to your degree? model, three dimensional dynamical system, in which variables are X, X' and theta. And then this three dimensional dynamical system which will be the preferred model for a forced oscillator in all cases is then chopped by a strobe light in the phase let's say fee = 0 and observed only there. So, upon forcing, we find a self-sustained oscillation. Which is represented by a point here. And the strobe section, that's every time you strobe this you see the oscillator in the same point. But if you strobe at different phases you see that this point actually moves around in a circle and comes back again and while moving around this way, also all of these points will then forget to interphase at zero we would have a cycle here. So this is the standard representation and since, as I said, the isochronous principle of Galileo each pendulum has a favorite period or frequency. Under weak-forcing you get the best result, that is to say, the largest amplitude of this resulting self-sustained oscillation of the forced system. ??? the forcing oscillator is going at the correct frequency, namely the natural frequency of the pendulum. But, suppose it's going in the wrong frequency. ??? experimental physics here. We have to try this out at home when no one is looking, with your own pendulum. (end of this side of tape). ...frequency, the natural frequency of this particular pendulum depending on the weight of the clock and the length of the string. And then if you force it, I'm forcing it but you can't see it, but I move my hand back and forth only, hopefully, only and at the same frequency. And my hand is moving to the left when the watch is moving to the right. It just happens to be the phase relationship that works here. Now, I'm forcing the natural frequency and I'm maintaining a self-sustained oscillation at that frequency. It's moving a lot faster, then we see that the watch does oscillate but that the amplitude is very small. This is the subject of the experimental work by Duffing where he obtained the following result: ??? watch. This entire phase portrait as a function of the frequency of the forcing oscillator that would be producing ? the larger pendulum and for me my hand then we obtain at/as? the favored frequency the larger oscillation and elsewhere the smaller one. So he did many experiments and this resulted after passing the favored frequency there is a reduction in again to a smaller. ??? frequency to slow, it's a small oscillation but at the same frequency, ??? in between a maximum, so this is a non-linear version of lessons? So what happens is ??? maximum, this one, and vanishes while this smaller one sort of stays small. So there are here two full bifurcations and that means you can almost observe of this- if you do it yourself - that as a natural frequency you have an oscillation of this magnitude and then you go faster and faster and then you slow down again at some frequency the amplitude of this oscillator increases enormously (through/to catastrophic chop?). So small oscillation suddenly jumps up to big one when this lesser bifurcation point is passed. So this is an application of catastrophe theory in the context of forced oscillation ???? (something about a zenith?) bifurcation. But Duffing discovered this, it's even provided in a mathematical model and this is the key example of a coupling ??? behaving in a forced oscillator ???. Now, signs bearing the frequency of the forced oscillator once could also determine the amplitude and I'm going to move on to the case of strong forcing so it would be helpful to have a control space in which we have this frequency as before but also amplitude of the forcing oscillator and then again we'll force this. I'll just describe the results. All of this that I described under (main roofing?) applies to the self-sustained oscillation of composite systems at approximately the natural frequency of the original system- the pendulum. Which is called the isochronous harmonic. So this anti-bifurcation? diagram deals with the so-called isochronous harmonic. One, one. Normal resonance. However, when you increase this frequency still more til you get twice the resonant frequency then it oscillates again, but goes around twice. So these can be indicated over here, here's the one to one isochronous that's this one. And then later on you have two to one is another. This means that in this region you'll observe the harmonic and outside this region you won't. .LP ?? these two points ??? zenith double hole takes place inside this tunnel, this zone, in which the one that winds harmonic is observed. And likewise there's double hole in two to one. Then there's three to one and so on. And then there's a half of this. One to two. Subharmonics, superharmonics, in fact at every rational number there's one of these tunnels. And furthermore, (let's now or I have) increase(d) the amplitude beyond this region where these simple observations are made and then, or course when the forcing oscillator is very, very strong, it is going to force the subject system to move at that frequency no matter what. So the if the amplitude of the forcing oscillation is great then the ??? region where it's trying to take place is going to be great. So these are constantly, indefinitely, spreading out. And likewise this one. And eventually they cross. And, after crossing, one observes in here, original chaos. So that's how to find chaos in these systems if you have an analog computer. We've been waiting for the day in which digital computers will catch up with the speed of analog computers but that's sort of happening this year. Now, the way in which these chaotic states appear in the forced oscillator context is- varies. Let me try to describe at least one. I was afraid of this, my time is about up. But we will ??/ chaos and then I'll just say a couple of words in summary about the term chaostrophe. It will take another talk I suppose do justice. So, I want to draw just one example of the Poincare cross section of forced oscillating. And consider the case in the Duffing situation where we have the isochronous harmonic inside the cusp there are two attractive states of different amplitudes of the same frequency and they each have their base which is separated by a curve ??? inset the saddle periodic attractor of- periodic orbit of ??? type. So here we have the larger or the normal puberty and we here? have the smaller orbit- amenoria - oscillators. And this is the strobe section of the ring. Now we have to enlarge this picture enormously to see what happens next, so I'll make this smaller and put it inside a bigger one. So we have two attractors and the saddle. And we did (or need?) some change to get these attractors farther away from the saddle. There's also a comp set (offset/opposite?) that goes with this attractor and an offset to another attractor. So what happens is called a normal/global? bifurcation under sufficiently strong forcing then this which is ??? curves around, crossing this which is the offset/opposite. Now, this cannot happen in a floral?, you know in a (Bankerfield?) dynamical system ???. You have to remember that this is a cross-section of a ring. And that all the trajectories are going out this way and going around and coming back. So the next instance of this point goes to ??? and comes back. It might be here. So since this is the inset, it has to be the contained, the future trajectory of this point. This point is on the inset of this saddle. So that means that this trajectory will spiral around, getting closer and closer to this saddle. But in backwards time, sorry this is now - I'm sorry. Even/given? if this point is here. And therefore this offset- it also would vary- it has to go through to here and there results this huge tangle discovered by Poincare. The third volume of my series is entirely about the geometry of these homoclidic? tangles. Anyway, in short, the simple system like the Duffing forced pendulum, where there are the two states and the saddle, anywhere there's a saddle, by increasing the force it can very likely result in this inset and outset apparatus of the periodic trajectory of saddlecutting? which is experimentally now observable get confused, intercept, and create this tangle. Which is fractal set. And this is chaos. This is the ??? of chaos and almost all the (or good?) cases f the forced oscillator ???. And ostensibly (or extensively?) studied- turned into alpha? as well as dynamic ??? but Smale ??? in the 1960s. So there is great literature ???. And then studied in great detail by experimentalists, particularly Ueda and his group in Japan. Ueda was a student of Hiyachi and a great pioneer. .RESUME EDITING Now, just one word about chaostrophe. All of this theory shows somehow we can make simple mathematical models to obtain complicated behavior, and therefore we can try to make simple models for complicated behavior in nature. Of course, if anybody tries to do that, there are immediate objections that the model is too simple. No matter if it explains a complicated behavior ??? - the math model is too simple. We want better models. But there is evidence, or at least there's reason to suspect that in the long run, in the future history of these scientific subjects, that simple models will actually prevail as excellent. I don't have time to explain it, and if I did you wouldn't believe me anyway. ??? It's possible one should not fire these people, or stand in the way of their promotions or try to block research in simple models because they may be a lot more positive than we suspect. In case you don't believe that, you can always start here and go backward. There's extensive subject[?] now of chaoscopy which has no literature - nothing is written down, there are few reports in the journals and yet there are a lot of very sophisticated programs, ??? computers, within the community. The persons in these programs do it with chaotic data is ??? very different ways. Some of them would be quite familiar to those of you involved in kinds of (serious analysis?) now such as the singular spectrum. Others are completely demented such as the ?????. It results in an estimate of a model which is adequate to predict that data. For example, if you take this chaotic attractor away here the horseshoe, it's fractal dimension is 2.1. Now we know that comes from a model and ??? and three dimensions. So we ought to get that the dimension is three. When we apply the fractal dimension algorithim to this data coming from three dimensions, maybe we just take one of those dimensions, say the x or the y or the theta, subject that to the algorithim and we find that the dimension of this attractor is 2.1. The theorem says you multiply by 2 you get 4.2, you add one, 5.2. If you add it up you get six. And in those dimensions of six you can actually reconstruct a model that would exactly predict this data. So, although the data looks very complicated, this algorithim of chaoscopy says you can model it exactly in six dimensions. Now, I ???? came from three dimensions. So the theory is not really that good. The fractal/final dimension of the universe has recently been measured in a master's thesis at UCSC and you ought to be happy to know that the dimension of the universe is 1.3. A little less than you might have expected but it's pretty sparse out there and that's exactly the same as the coastline of Denmark. So chaoscopy ??? it reconstructs the whole picture like this and it's possible to get equations, for those who like equations- I don't think they're so important myself, except when you have no other means, you can then get equations defining a mathematical model ??? usual language of applied mathematics which if you then- lets you study through simulation on analog or digital computers you would then get the data you got exactly, but furthermore you could forecast it for any time into the future. So these forecasting methods so far have not been shown to be really better than the best methods used by my nomineer?, you know the traditional nomineers? ???. That's the surveillance/survey methods?. .LP end of lecture beginning of audience comments and questions.