Visualisation and dynamical systems


John Hubbard



When asked to explain what my mathematics is about, I often answer by showing pictures illustrating the behavior of the dynamical systems I study. Non-mathematicians usually respond with interest, at least polite interest but sometimes much more. They often express amazement that my pictures are in any way related to mathematics. Mathematicians are also often interested, but many (fewer as time goes on) dismiss what I do as some sort of “fad math" devoid of theorems.

Visualization can be essential even if the object is the purest of mathematics. The pictures generated by computers using non-linear dynamical systems may or may not be beautiful; they may or may not be art; there doesn't seem to be a \true" answer to such questions, and each viewer  must come to his or her own conclusions. But the pictures most definitely contain information of essential interest to mathematicians.

They help us to formulate conjectures, which in many cases would be inconceivable without the pictures. Experimentation, again using computer graphics, often can guide us to a proof, from which any computer aspect is totally absent. Finally, in the essential task of communicating our discoveries to others, the computer graphics may be absolutely essential.


Complex dynamics and visualization


I remember Lars Ahlfors, in 1982, telling me that in his youth his adviser Ernst Lindelöf made him read the memoirs of Fatou and Julia on the iteration of rational functions. These memoirs, he told me, struck him at the time as “the pits of complex analysis." He said that he only understood what they were about when seeing the pictures Mandelbrot and I were showing. If even Ahlfors, the creator of some of the principal tools in the field, couldn't see what the authors were getting at, what of lesser mortals? Indeed, those memoirs were practically forgotten for 60 years, waiting for computer graphics to reveal what Fatou and Julia had glimpsed.


The dynamics of the pendulum


I will now present a more personal example: the movements of the forced damped pendulum, governed by the \garden-variety" differential equation


q’’ + a q  + b sin q = c cos(wt):


Since a robot is an assemblage of forced damped oscillators, it is of great interest to understand the dynamical properties of such an object. Moreover, this differential equation has been studied by generations of students in courses in ordinary differential equations, either as an example in perturbation theory (develop the solutions in power series with respect to b, since the equation is linear when b = 0) or an example to test various numerical methods. Apparently, none of these studies led to any precise understanding of the behavior of solutions to the equation.

A bit of computer investigation for the values a = .1; b = c = w = 1 leads to the observation that these exists an attracting oscillation of the system of period T = 2p, corresponding to the downward equilibrium of the unforced pendulum. But the pendulum can get from an initial state to the attracting oscillation in many ways, and one essential measurement of the difference between two such evolutions is how many times the pendulum \goes over the top" before settling down. Certainly if two evolutions correspond to different numbers of such passages, they must be quite uncorrelated. If we color the plane of initial states (positions and velocities) according to how many times the pendulum goes over the top before settling down, we obtain the following picture.







FIGURE 1. The plane of intial states colored according to the number of times the pendulum goes over the top before settling down.


Contemplating this picture led to the conjecture that the basins form Lakes of Wada : every point in the boundary of one is in the boundary of all the others. When originally discovered by Brouwer and Yoneyama [4], this sort of behavior was seen as pathological; I am sure neither thought that such things would show up in mathematical problems of an applied nature.

Drawing the stable and unstable manifolds of the unstable periodic solution corresponding to the upper equilibrium of the pendulum leads to another conjecture: by choosing the initial condition correctly, the pendulum can be induced to go through any sequence of gyrations one wants, for instance turning once counterclockwise, then three times clockwise, then spending time almost vertical, then turning 5 billion times lockwise, and then once counterclockwise, etc.






FIGURE 2. A quadrilateral in the plane of initial conditions, with its forward and backward images going through itself three times, forming a Smale Horseshoe. The quadrilateral is “fitted" to the unstable manifold of an unstable equilibrium.




In both cases, with the help of the computer, the conjectures can be proved, using techniques due to Yorke, Kennedy and Nusse [2] for the first, and to Smale [3] for the second. The details are given in [1].


Some conclusions


This story has implications, for mathematics, for science and engineering, and for education.

In mathematics, the use of visualization leads to interesting conjectures, which would never have been contemplated without such technology. The entire field of complex dynamics is filled with similar examples: I am convinced that without computer graphics, the field would simply not exist, and I know that in the parts I have participated in, the motivation provided by computer graphics was essential, even if computers are never mentioned in the proofs.

Further, as Fatou and Julia discovered, even if you can prove theorems, you often can't communicate why they are interesting without the illustrations provided by computers.

The implications for science are probably more important yet. It is clear, from looking at the pictures and from the proofs, that the motion of a pendulum is unstable and chaotic: the challenge is to harness this instability to make robots more efficient. One can imagine two scenarios: a robot moving awkwardly, frequently stopping to reset its position to within prescribed tolerances before going on to the next task, like a beginning skier who stops between turns to regain his balance. But exploiting the instabilities should lead to the robot moving fluidly, expending far less energy, like an experienced skier floating down the slope, constantly out of balance but with effortless ease.

At the moment, robots move according to the first scenario; they would be immensely more effective if we could move them to the second.

Where teaching of mathematics is concerned, I have found that computers can be immensely successful in bringing topics to life, and any teacher of math can attest to how essential this is: the mathematical objects students are expected to study often, in fact usually, have no reality in the student's minds, which prevents them from thinking about them effectively. Computers also allow the use of far more interesting examples, much closer to real problems. The pendulum example has been used in several undergraduate classes; this would have been inconceivable without the computer.




[1] J. H. Hubbard, The Forced Damped Pendulum: Chaos, Complication and Control, American Mathematical Monthly, 106, 8, (1999), pp. 741-758.

[2] J. Kennedy and J. Yorke, Basins of Wada, Physica D 51 (1991), 213-255.

[3] S. Smale, Diffeomorphisms with many periodic points, Differential and combinatorial topology, edited by S. S. Cairns, Princeton U. Press (1965) 63-80.

[4] K. Yoneyama, Theory of continuous set of points, Tohoku Math. J., 11-12 (1917), 43.