Invitation to a Contemporary Physics (2004)

calculator. Select some value of the control parameter ‘b.’ Enter then a seed value X0 as input on the right-hand side of the logistic equation, evaluate the expression bX0(1 −X0) and just output it as X1. To get X2, we have to use the just evaluated X1, as the new input and out comes X2. The procedure can be iterated n times to get Xn at the nth round, and so on for the whole sequence. Now repeat this numerical exercise with another value of the seed X0, and look for any change in the pattern of the sequence of numbers that come out. (Note here that our X’s by definition lie between zero and one. It is readily verified that this constrains the control parameter b to lie between zero and four.) Is the pattern periodic? Does the period change with b? Or otherwise, is there a pattern at all? These are the questions to be answered. We are zeroing on chaos.

All this can be viewed live on the screen of your PC with the help of a few lines of statements in BASIC. But, for a clearer understanding of what is really going on, it is best to resort to the following graphical construction. Just plot Xn+1 (vertically) against Xn (horizontally) using them as the Cartesian coordinates. This is our phase space if you like: a finite phase space, a unit square. On this plot, our logistic function bXn(1 −Xn) against Xn is nothing but a parabola standing on the unit horizontal base (Fig. 7.5).

Now start with the input seed X0 marked on the horizontal axis. To get X1, just move vertically up (or down as the case may be) meeting the parabola at the point P0 on it. It is clear that P0 has the coordinates (X0, X1). In order to get X2 now, we have to use X1 as the new input on the horizontal axis and repeat the above procedure. It is much more convenient, however, to draw a 45◦ straight line, diagonally across the phase space on which Xn = Xn+1. With this, to get X2 from X1, all we have to do is to move horizontally from the point P0 across up to the diagonal and then move vertically meeting the parabola at a point P1, say. Clearly, P1 will have the coordinates (X1, X2). The procedure can now be iterated to generate the whole sequence, X0, X1, . . . , Xn, Xn+1, . . . , ad infinitum (Fig. 7.5).

7.3.3 The Period Doubling Bifurcation

With the graphical construction in hand, we can at once make several observations by mere inspection. Thus, for su ciently small values of the control parameter b, the parabola lies entirely below the diagonal line. It is readily seen that starting with any seed value X0 whatsoever, we quickly cascade down and finally converge to the origin (Fig. 7.5a). This makes the origin an attractor. Let us denote this point by P0 and the corresponding X value by X0 (= 0, of course). It is clear that on further iteration X0 → X0 . Hence it is a fixed point, in fact a stable fixed point. That is, starting with any X0, we end up there.

This situation persists as we increase our control parameter b until the diagonal is just tangential to the parabola at the origin. This threshold value, b1 say, of the control parameter can be obtained in a straightforward manner by equating the

X1, . . . , Xn, Xn+1, . . . fails to converge to any fixed point at all. Instead it settles

down to a 2-point periodic cycle X2 → X3 → X2 → X3 → · · · . We call X2 and X3 elements of the cycle (Fig. 7.5c). We say that the earlier fixed point X1

has bifurcated to the alternate pair (X2 , X3 ) that forms an attractor of period 2, or a 2-cycle. This is the famous period doubling. The alternating sequence X2 → X3 → X2 → X3 → · · · is reminiscent of the pair of alternating long and short intervals of the dripping faucet discussed in the last section. The general trend should be clear now. At the next threshold value b2, say, of the control

parameter b, the 2-cycle in turn becomes unstable and we have instead a 4-point periodic cycle X3 → X4 → X5 → X6 → X3 → X4 → X5 → X6 → · · · . So we now have a 22-cycle. It is simply that each element of the 2-cycle has bifurcated to

two elements (see bifurcation plot in Fig. 7.5c). And so on to the 23-cycle, and in general to the 2k-cycle at the kth threshold bk. What you get looks like a cobweb plot for the logistic map. It turns out that the successive thresholds of instability come on faster and faster and converge to a point of accumulation b∞ as k → ∞, where we have a 2∞-cycle: the period becomes infinite! The pattern never repeats itself. It has become aperiodic. This is the onset of chaos, and the entire sequence is known as the Feigenbaum scenario of the period-doubling bifurcation route to chaos. The critical value of the control parameter for the onset of chaos turns out to be b∞ = 3.569 . . . , the Feigenbaum constant.

7.3.4 Universality

The approach to criticality is subtle and interesting. The ratio of the successive intervals between the threshold values of the control parameter, the so-called bifurcation ratio (bk − bk−1)/(bk+1 − bk) tends to a limiting number δ = 4.669 · · ·

as k tends to infinity. The entire sequence of events leading up to chaos seems to simulate the dripping patterns of the leaking faucet surprisingly well. Could this be a mere coincidence? Well, it could have been suspected to be so but for the fact that the behavior observed for the logistic equation actually turns out to be universal within a whole class. As was shown by Feigenbaum, we can replace the parabolic (quadratic) map by any other single-hump map, but one with a smooth maximum, without changing the period doubling bifurcation sequence or the associated limiting ratio δ and other critical scaling exponents characterizing the onset of chaos. On the other hand, replacing the parabola with a ‘tent’ having a sharp triangular apex at the maximum point is a di erent matter. This would be in a di erent class. But Xn+1 = r sin(πXn) belongs to the same class as the logistic map. This is something familiar from the modern theory of second-order phase transitions due to Kenneth G. Wilson. It is this universality of the critical behavior that makes an algorithm as simple-looking as the logistic map capture the essential physics of diverse systems close to a crisis. After all, it is the crisis that brings out the intrinsic character common to a whole class of individuals.

previous section. This is what gives the sensitive dependence on initial conditions. It can be shown that under this stretching and folding, the Θn values hop over the entire unit interval at random, eventually covering it densely. The entire unit interval is the attractor — the strange attractor, as we will see later. This is fully developed chaos! Similar things happen in the range b∞ < b < 4 also. However, here the Θn values hop randomly on a subset of the unit interval whose dimension is less than unity, as we will see later: it is a fractal (see Section 7.5). Indeed, at the onset of chaos (b∞ = 3.569), the fractal dimension of the strange attractor can be shown to be 0.537. For b = 4, however, the dimension is actually an integer (= 1), and Xn covers the entire unit interval densely.

The logistic map is a kind of ‘Feigenbaum Laboratory’ in which we can do numerical experiments with chaos and learn a great deal. One wonders, however, how such a discrete map can approximate reality when all real processes happen in continuous time. We address this question next.

7.3.6Poincare´ Sections: From Continuous Flows to Discrete Maps

A discrete map of the kind we have discussed above can be constructed for a real dynamical system quite naturally by taking the so-called Poincar´e section of the continuous flow, or the trajectories, in the phase space of the system. Figure 7.7 shows such a Poincar´e section for a dynamical system having a three-dimensional phase space. The surface of section has been taken to be a plane perpendicular to the z-axis. But other choices are possible. Here, instead of watching the trajectory continuously in time, we simply record the sequence of the points P0, P1, . . . of intersection of the trajectory with the chosen plane as the trajectory crosses it in a given direction — of the negative z-axis, say. In e ect we have replaced the continuous phase flow in time described by the di erential equation, by a discrete mapping relating the (x, y)-coordinates of the successive points Pn, Pn+1 of intersection — an algebraic, di erence equation. We may have, for example Xn+1 = 1 − cXn2 + Yn

z

S

y

x

Figure 7.7: Poincar´e section (P0, P1, . . .) of a phase space trajectory by the plane S.

and Yn+1 = bXn, where ‘c’ and ‘b’ are adjustable parameters, e.g., we may have c = 1.4 and b = 0.3. This two-dimensional discrete map, called the H´enon map after the French astronomer Michel H´enon, describes the Poincar´e section of a chaotic attractor in a three-dimensional phase space, of which more later. The Poincar´e section of a three-dimensional flow will be, in general, a two-dimensional surface. But if the dissipation is large and the areas contract rapidly, then the section will consist of points distributed along a curve. In other words, we now have a onedimensional map Xt → Xt+1. This is often called the first-return map.

Taking Poincar´e sections is a very revealing technique for studying, or rather visualizing the geometrical forms, such as the orbits and their limits in the phase space (the phase portraits). It certainly reduces the dimension by one. And yet it retains the all important qualitative, global features of the dynamics, e.g., the divergence or convergence of neighboring trajectories, expansion or contraction of phase-space volume elements, the periodicity or aperiodicity of the orbits and many other signatures of dynamics. Also, the algebraic di erence equations are easier to handle than the original di erential equations. One can, for example easily implement the H´enon map on a programmable calculator. The Poincar´e section is particularly useful for the study of attractors in higher dimensions. We will look at these attractors next.

7.4 Strange Attractors and Routes to Chaos

Attractors are geometric forms in the phase space to which the phase trajectories of the dynamical system converge, or are attracted and on which they eventually settle down, quite independently of the initial conditions. The idea of an attractor is quite simple but very powerful for a global qualitative understanding of the motion, both regular and irregular, without having to solve the equations of motion — which is seldom possible anyway. It is best illustrated through examples.

7.4.1 Stable Fixed Point

The simplest attractor is a fixed point, or rather a stable fixed point. Consider the phase portrait of a damped linear oscillator, a pendulum with friction for example. The phase space is two-dimensional comprising the velocity (momentum) and the position coordinates. Because of damping the phase point spirals in onto the origin and rests there (Fig. 7.8a).

The origin is thus a fixed point. It is obviously stable — displaced from it, the system returns to it eventually. Also, all trajectories, no matter where they begin, are attracted towards it — hence the name attractor. It is also readily seen that any element of the phase space will contract in its extension as it is attracted towards the fixed point. As there is contraction along the two independent directions in the two-dimensional phase space, both the Lyapunov exponents are negative. This

‘contraction’ is the meaning of the term ‘dissipative flow’ in phase space language. The most interesting flows in Nature are dissipative and have to be maintained by external driving forces, such as stirring, heating, pumping, kicking, etc. We call them open systems.

7.4.2Limit Cycle

Next comes the limit cycle. It is a closed loop in the phase space to which the trajectories converge eventually (Fig. 7.8b). The limit cycle corresponds to a stable oscillation. Here, one Lyapunov exponent is zero (along the loop) and the rest are negative. Its Poincar´e section is just a point. Again, the flow is dissipative. For a two-dimensional phase space, the limit cycle is the only attractor possible, aside from the fixed point. This is a direct consequence of the condition that the phase trajectories cannot have self-intersection. To have anything more complicated, the trajectory must escape in a third dimension. (Recall that from a given point a unique trajectory must pass.) The limit cycle is at the heart of the most simple periodic processes in Nature — the beating of the heart, the ‘circadian rhythms’ of period 23 to 25 hours in humans and animals, the cyclic fluctuation of populations of competing species in an ecosystem, oscillating chemical reactions like the Beluzov– Zhabotinsky reaction, marked by colour changes every minute or so, etc. The limit cycle thus is a natural clock.

7.4.3 The Biperiodic Torus

The next most complicated attractor has the geometric form of a doughnut or anchor ring — the torus (Fig. 7.8c). The trajectories converge on the surface of the torus, winding in small circuits around the axis of the torus (at a frequency f2) while orbiting in large circles along the axis (at a frequency f1). Here, two of the Lyapunov exponents are zero and the rest are negative. This corresponds physically to a compound or biperiodic oscillation resulting from the superposition of two independent motions. If the frequencies f1 and f2 are commensurate, that is if the ratio f1/f2 can be expressed as the ratio of two integers, then the trajectory on the torus closes on itself — frequency locking. The motion is actually periodic then. A Poincar´e section of the trajectory will be a finite set of points traversed by the successive go-rounds. If, on the other hand, the frequencies f1 and f2 are incommensurate, the trajectory will cover the torus densely and the Poincar´e section will be a continuous closed curve. The motion will be quasiperiodic. Such a biperiodic torus attractor is known to show up in the Couette flow. But the simplest example will be two coupled oscillators of di erent frequencies. Incidentally, if the frequencies are close enough and the coupling is nonlinear, the oscillators may get locked into a common frequency mode. This is called ‘frequency entrainment,’ lockin, noted first by the great 17th century Dutch physicist Christiaan Huygens. He

dissipative flows which are regular, that is, stable and predictable to any degree of accuracy. These flows are essentially periodic. There is, however, an entirely di erent type of low-dimensional attractor that characterizes flows which are irregular, that is, unstable and unpredictable in the long term. Such an attractor is a phase-space non-filling set of points or orbits to which all trajectories from the outside converge, but on which neighboring trajectories diverge. Thus, at least one Lyapunov exponent has to be positive. It is ‘strange’ in respect of the geometry of its form as well as in terms of its manner of traversal of this finite region of phase space. It is called the strange attractor and is the engine that drives chaos. It was discovered jointly by the Belgian mathematician David Ruelle and the Dutch mathematician Floris Takens around 1971.

The strange attractor is the answer to the question we had posed earlier: how can a trajectory remain confined forever to a finite region of phase space, without ever intersecting itself as it must not, and without closing on itself, as again it must not for that would mean a periodic motion? To make matters worse, the system happens to be dissipative — eventually the phase-space volume must contract to zero. You see the conflict! Just think of a one-meter long strand of the contortionist DNA packed within a cell measuring one millionth of a meter across, and you will get some idea of the packing problem, which is actually infinitely worse. The conflict of demands for zero volume and eternal self-avoidance is resolved by making the attractor into a fractal — a geometric form with fractional dimension lower than the phase space in which it is embedded. Fractals are an ingenious way of having surface without volume if you like. We will now illustrate all this with two celebrated examples of the strange attractor: the H´enon attractor of the discrete two-dimensional H´enon map for its simplicity, and the Lorenz attractor of the three-dimensional dimensional convective flow for its complexity. The latter is also historically the first known example of a strange attractor.

7.4.5The Henon´ Attractor

As already mentioned, the H´enon map, discovered by H´enon in 1976, is given by Xn+1 = 1 − cXn2 + Yn and Yn+1 = bXn. Thus, starting with any point in the (X, Y )-plane, one can generate the entire attractor by iterating the map a large number (usually 105–106) of times. It can be shown that the map is dissipative for a magnitude of b less than unity, i.e., for |b| < 1. (It is conservative (phase volume preserving) for |b| = 1.) For b = 0, the dissipation is so large that the map contracts to a one-dimensional quadratic map of the kind we have discussed earlier. For a convenient value of b = 0.3 (moderate dissipation), we can now study the map as a function of the control parameter c. The map follows a period doubling route to chaos. For c = 1.4 we have a strange attractor, shown schematically in Fig. 7.9.

Two crucial features are to be noted here. One of these has to do with the geometry and the other with the motion. First, the attractor has a self-similar