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Once set in motion, this double pendulum moves under the force of gravity in an extremely intricate manner. As it gradually loses energy due to slight friction, the type of motion changes. It is particularly hard to define in an intermediate energy range. The pendulum sometimes turns over once more . . . and sometimes pendulum here. It seems hardly falls back, like the inner possible to make a prediction. Some idealisation is of course necessary to describe motion as complex as this. The effect of friction will for example be completely disregarded in the following mathematical description of the double pendulum. The two masses will also be presumed to be concentrated in points in such a way that the outer pendulum is suspended in the mass point of the inner. For the sake of simplicity the two pendulums are assumed equally long and their masses equally heavy. The configuration of the double pendulum will be described by two angles, phi 1, for the inner pendulum, and, relative to this, phi 2, for the outer. This mathematical model will be simulated an the computer. First, we show in three computer experiments the relationship between motion and energy. E = 0.5 E= 5 E = 50 We choose typical values in the low, middle and high energy ranges. At the low energy value of 0.5 the two pendulums exhibit small oscillations around their stable equilibria. Energy 5 is sufficient to bring the pendulums into rotation around their respective Suspension points. At energy E = 50 the two pendulums move almost as if there were no force of gravity. In this case the angular momentum is a second conserved quantity besides the energy (E).Types of motion: Periodic Quasiperiodic Chaotic The motion can be periodic,

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quasiperiodic or chaotic. While the actual motion is taking place

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an the left-hand side, we record the trace of the outer mass point an the right. This record enables us to make comparisons with earlier configurations, and we may distinguish between various types of motion. In the case of periodic motion the trace closes upon itself. In the case of quasiperiodic motion the trace looks almost as regular as periodic motion; the difference is that the trace is not closed.

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If we follow this motion for a while, we See that its trace covers the plane with pretty good regularity. The double pendulum never returns to exactly the Same state in which it was previously. The motion is therefore not periodic. However, in regular intervals it returns to a close neighbourhood of its starting point

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- hence the term "quasiperiodic". Chaotic motion is very different in that it leaves behind a fairly irregular trace. Look carefully here at the way the outer pendulum sometimes rotates through and sometimes swings back. Over and over again it is forced to decide between these two alternatives, and its choices are by no means regular. Clearly, Small differences in the initial conditions have major consequences later on. Here we

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See two double pendulums starting under almost identical initial conditions. For a while their traces can hardly be distinguished. Then suddenly their ways part: at the top, the outer pendulum chooses to swing back, while at the bottom it rotates

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through. From now an there is hardly any similarity between their motions. Let us repeat the critical phase. This is where the paths of the two pendulums separate. HENRI POINCARÉ had an idea how the complexity of this motion might be visualized. He suggested a strategy to record only particular moments rather than the complete process. For example one could always capture the Situation where the double pendulum goes through the stretched configuration with a positive angular velocity of the outer pendulum .... . . like now . . . now . . . and now. At such moments the angle phi 1 of the inner pendulum and its corresponding angular momentum should be measured and one plotted versus the other. In this way one would obtain a point of the so-called phase-plane of the inner pendulum. While the motion progresses, one waits for the return of such stretched configurations and records them in the phase-plane, point by point. Generation of a Poincaré Section The large area an the right is the phase-plane of the inner pendulum, in which the angle cp 1 varies along the abscissa while the corresponding angular momentum is given an the ordinate. We choose an arbitrary initial point and follow the motion until the distinguished stretched configuration has again been reached . . .. . . now! . . . and now . . . and now. Each time we measure the angle phi 1 and the corresponding angular momentum and thereby obtain a new point in the phase-plane. At the bottom left the trace of this motion is recorded. However, from now an we plot it in a different manner than before: The angle phi 1 of the inner pendulum is given horizontally while the vertical axis represents the angle phi 2 of the outer pendulum. After a while the sequence of these points condenses into a line: this is typical of quasiperiodic motion. The elliptical form of this line indicates resonance in the centre. When we choose the centre of the ellipse as a starting point, the outer pendulum rotates with exactly the same frequency as the inner pendulum oscillates. The motion

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is periodic; the same point appears again in the Poincaré section. Now we take a third initial condition, and again follow in detail the generation of the subsequent points. They finally form a line that crosses through the phase-plane. The motion is quasiperiodic. The fourth initial condition produces a less regular type of motion. This can already be seen in the bottom left trace.After the

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first few points have been generated, the outer pendulum changes its direction of rotation, and for a while it passes the stretched configuration at negative angular velocities only.There are therefore no points in the Poincaré section. Only now do we see the reappearance of points. In contrast to the previous cases, however, they are not confined to a line but are distributed over an area which in the long run they would fill densely.This type of motion is called chaotic; the points in the Poincaré section fill up a so-called "chaotic band".We calculate 300 successive points for each of a number of further initial conditions - and finally obtain this picture. The Poincaré sections give an indication of the rich

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complexity in the dynamics of a double pendulum of this sort. They condense in one picture the long-term behaviour of a large number of trajectories, in a way that the eye can easily follow. They show how closely interwoven - at one and the same energy value - regular and chaotic behaviour can be. A small change in the initial condition - and the features of the eventual trace may be quite different. Under such circumstances long-term prognosis becomes practically impossible. The system's behaviour is in certain respects unpredictable, although of course, it is strictly determined by the equations of motion. In order to get an overall view, let us now consider the Poincaré sections at twelve different energy levels. Poincaré Sections at Different Energies Let us begin with the low energy 0.5. The white line encircles the phase-plane area that is accessible at this energy. The Poincaré section consists essentially of lines. It contains two major resonances, which correspond to the two oscillating eigenmodes. At the central resonance the two pendulums oscillate out of phase. Right at the top of the Poincaré section is the resonance that corresponds to the second eigenmode. Both pendulums oscillate in phase here. The resonance we are now following can be viewed as a combination of the two eigenmodes. The line structure of this picture reflects the regularity of

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the motions at this low energy. When the energy is doubled, some of these lines have broken up into chains of islands. At the energy 1.5, chaos is clearly visible.

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The central resonance has split into two resonances. Chaos begins to dominate, and little room is left for the islands of order. It is interesting

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to see how the central resonance has developed after the bifurcation. We first observe the trace of the left-hand resonance: it has the shape of an ellipse and evolves in a clockwise manner. The right-hand resonance produces the same trace, but this time it evolves in the opposite direction. The two resonances are therefore images of one another under time reversal. At energy four we find nothing but chaos. As the energy is raised further, order emerges again from the chaos, at first in the

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form of the central resonance that we are now going to observe: the outer pendulum

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completes a rotation at exactly the same frequency at which the inner one oscillates. At further increasing energy the ordered area seems to shrink again.But regular motion finally takes hold and also appears at other places in the phaseplane .

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. .. . . as for instance around this resonance, which consists of two points in the Poincaré section. Here we show the course of

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the corresponding motion. At energy ten we find for the first time that a line crosses all through the phase-plane and divides the chaos into two parts, an upper and a lower one, between which the motion can no longer mediate, even an a long-term basis. As energy is increased still further, more and more

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of these continuous lines appear, and the motion becomes more and more regular. Let us see what main resonances occur at high energies. We have already encountered the red resonance,

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so let us focus on some of the others. At this resonance the outer pendulum oscillates exactly once while the inner pendulum completes one rotation. This is the time reflected

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version of the resonance we have just seen. The traces are identical, but the motion progresses in opposite directions. We find another resonance hidden in a chaotic band. Here the two pendulums rotate in synchrony: the inner clockwise, the outer counter clockwise. At energy rising to infinity, or at vanishing gravity, the chaos disappears completely. The angular momentum becomes a conserved quantity, and the motion proceeds an lines that lie parallel to the cpl axis. W e describe it by noting how much to the right or left the points an the ordinate are displaced in one step of the Poincaré process. This is shown by the white lines that

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cross this picture. Let us follow the motion for some typical starting points that we pick an the ordinate. In the upper yellow area the inner pendulum rotates counter-clockwise, while the outer one oscillates. In the blue area both pendulums rotate counter-clockwise. Note how from point to point the angle phi 1 always increases by the same amount. This starting point is so chosen that the inner pendulum does not rotate but only oscillates while the outer one rotates. In the green area the two pendulums rotate counter to one another, so that the total angular momentum is small. Finally, in the orange-coloured area the motion, up to time reversal, is the same as in the yellow area. Decay of the Last KAM-Trajectory: The Golden RationLet us quickly recall the Poincaré sections as energy increases. At first, order dominated in the form of quasiperiodic motions. Then chaos spread and finally covered the whole phase-plane. Then islands of ordered motion reemerged, and here, at energy ten, a green belt appeared for the first time, crossing the phase-plane. In a little while we shall look at this boundary between two chaotic domains in greater detail. But first let us go an raising the energy. More and more separating lines appear, called KAM-lines for short after the mathematicians KOLMOGOROFF, ARNOLD and MOSER, who proved their existence. The final Poincaré section is composed of such KAM-lines only. From here on we are going to move backwards to follow the scenario of the decay of order as energy decreases. We know that one of the KAM-lines will be the last to survive. Let us therefore look at this section of the green area in close-up. The first thing we do is to look at the most important resonances. For this initial condition the angle cp2 rotates exactly twice as fast as

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phi 1.For this initial condition the ratio of the angular velocities is exactly 1:3. This resonance is therefore characterised by three points. Between the two resonances just shown, there lies a resonance of five points with a ratio of the angular velocities of 2:5 - this ratio is called the "winding number". Resonances always have rational winding numbers, while quasiperiodic motions have irrational winding numbers. Between the resonances with winding numbers 1:3 and 2:5 there lies another one with a winding number of 3:8. We begin to realize a simple algorithm. The two last winding numbers give us the next one, by simply adding numerator to numerator and denominator to denominator. This resonance has accordingly the winding number 5:13, the next one 8:21, and so on. The numbers rapidly approach a particular irrational number - the golden ratio. As we shall see, this irrational winding number characterizes the quasiperiodic motion whose KAM-line keeps chaos in check longest as energy is reduced. At energy 20 the resonances just described appear as conspicuous chains of islands, with green KAM-lines and chaotic bands beside them. At energy ten the resonances are still clearly visible,

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while the chaotic bands have spread and are now confined by one line only, namely

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the "last KAM-line". Let us pick a starting point an this KAM-line and observe the last stable quasiperiodic motion in the middle of chaos. The remarkable fact that the golden ratio characterizes the last ordered motion before complete takeover by chaos, is not only true for the double pendulum. It is one of the amazing universal features of complex dynamics that computer experimentation has brought to light in recent years. If we now lower the energy a little, the golden KAM-line decays - and with

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it the highest resonances. When we follow the motion of a starting point from somewhere near the decayed KAM-line, it seems at first to be pretty well ordered, but then develops in a way we have now learnt to identify as chaotic.