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However, it is likely that the results obtained for these difficult cases will be analogous to the preceding ones, at least in their degree of complexity. But, in astronomical problems, the situation is quite different: the constants defining the motion are only physically known, that is with some errors; their sizes get reduced along the progresses of our observing devices, but these errors can never completely vanish.
So far, the idea that some physical systems could be complicated and sensitive to small variations of the initial conditions—making predictions impossible in practice—remained hidden in very confidential mathematical papers known to a very small number of scientists. One should keep in mind that by the turn of the century, physics was triumphant and the general opinion was that Science would eventually explain everything.
The revolutionary idea that there is a strong conceptual limitation to predictability was simply unacceptable to most scientists.
Why is it that showers and even storms seem to come by chance, so that many people think it quite natural to pray for rain or fine weather, though they would consider it ridiculous to ask for an eclipse by prayer? We see that great disturbances are generally produced in regions where the atmosphere is in unstable equilibrium.
The meteorologists see very well that the equilibrium is unstable, that a cyclone will be formed somewhere, but exactly where they are not in a position to say; a tenth of a degree more or less at any given point, and the cyclone will burst here and not there, and extend its ravages over districts it would otherwise have spared. If they had been aware of this tenth of a degree they could have known it beforehand, but the observations were neither sufficiently comprehensive nor sufficiently precise, and that is the reason why it all seems due to the intervention of chance.
He was no longer considering chaos as an obstacle to a global understanding of the dynamics, at least from the probabilistic viewpoint. Unfortunately, none of his contemporaries could grasp the idea—or maybe he did not formulate it in a suitable way—and one had to wait for seventy years before the idea could be re-discovered! You are asking me to predict future phenomena. If, quite unluckily, I happened to know the laws of these phenomena, I could achieve this goal only at the price of inextricable computations, and should renounce to answer you; but since I am lucky enough to ignore these laws, I will answer you straight away.
And the most astonishing is that my answer will be correct. Another attempt to advertize these ideas outside mathematics and physics was made by Duhem in his book The aim and structure of physical theory.
On such a surface geodesics may show many different aspects. There are, first of all, geodesics which close on themselves. There are some also which are never infinitely distant from their starting point even though they never exactly pass through it again; some turn continually around the right horn, others around the left horn, or right ear, or left ear; others, more complicated, alternate, in accordance with certain rules, the turns they describe around one horn with the turns they describe around the other horn, or around one of the ears.
Finally, on the forehead of our bull with his unlimited horns and ears there will be geodesics going to infinity, some mounting the right horn, others mounting the left horn, and still others following the right or left ear.
But, for the physicist, this deduction is forever unutilizable. When, indeed, the data are no longer known geometrically, but are determined by physical procedures as precise as we may suppose, the question put remains and will always remain unanswered.
Unfortunately the time was not ripe. The theory went into a coma. However, meanwhile, physics and mathematics had gone through several revolutions and non-predictability had become an acceptable idea. More importantly, the world had also gone through several more important revolutions. This is probably the explanation of the success of the butterfly effect in popular culture. One should certainly mention the best seller Chaos: making a new science Gleick which was a finalist for the Pulitzer Prize.
One should not minimize the importance of such books. One should also emphasize that Lorenz himself published a wonderful popular book The essence of chaos in At first unnoticed, it eventually became one of the most cited papers in scientific literature more than 6, citations since and about each year in recent years.
For a few years, Lorenz had been studying simplified models describing the motion of the atmosphere in terms of ordinary differential equations depending on a small number of variables.
For instance, in he had described a system that can be explicitly solved using elliptic functions: solutions were quasiperiodic in time Lorenz His article Lorenz analyzes a differential equation in a space of dimension 12, in which he numerically detects a sensitive dependence to initial conditions. His paper lead him to fame. In this study we shall work with systems of deterministic equations which are idealizations of hydrodynamical systems.
After all, the atmosphere is made of finitely many particles, so one indeed needs to solve an ordinary differential equation in a huge dimensional space. Of course, such equations are intractable, and one must treat them as partial differential equations. In turn, the latter must be discretized on a finite grid, leading to new ordinary differential equations depending on fewer variables, and probably more useful than the original ones.
Another bibliographic reference is a book by Birkhoff on dynamical systems. A missed opportunity? Lorenz considers the phenomenon of convection. A thin layer of a viscous fluid is placed between two horizontal planes, set at two different temperatures, and one wants to describe the resulting motion.
The higher parts of the fluid are colder, therefore denser; they have thus a tendency to go down due to gravity, and are then heated when they reach the lower regions. The resulting circulation of the fluid is complex. Assuming the solutions are periodic in space, expanding in Fourier series and truncating these series to keep only a small number of terms, Salzman had just obtained an ordinary differential equation describing the evolution.
Obviously, one should not seek in this equation a faithful representation of the physical phenomenon. It was a good choice, and these values remain traditional today. He could then numerically solve these equations, and observe a few trajectories. The anecdote is well known Lorenz : I started the computer again and went out for a cup of coffee. When I returned about an hour later, after the computer had generated about two months of data, I found that the new solution did not agree with the original one.
Let us introduce some basic terminology and notation. The purpose of the theory of dynamical systems is to understand the asymptotic behavior of these trajectories when t tends to infinity. He notices that these unstable trajectories seem to accumulate on a complicated compact set, which is itself insensitive to initial conditions and he describes this limit set in a remarkably precise way.
There exists some compact set K in the ball such that for almost every initial condition x, the trajectory of x accumulates precisely on K.
Thus within the limits of accuracy of the printed values, the trajectory is confined to a pair of surfaces which appear to merge in the lower portion. The trajectory turns around the two holes, left or right, in a seemingly random way.
This defines a two to one map from the interval to itself. Indeed, in order to go back in time and track the past trajectory of a point in [0,1], one should be able to select one of the two surfaces attached to the interval. Numerically, the first return map is featured on the left part of Figure, extracted from the original paper.
In particular, the periodic points of f are exactly the rational numbers with odd denominators, which are dense in [0,1].
What an intuition! Finally, he concludes with a lucid question on the relevance of his model for the atmosphere. There remains the question as to whether our results really apply to the atmosphere. One does not usually regard the atmosphere as either deterministic or finite, and the lack of periodicity is not a mathematical certainty, since the atmosphere has not been observed forever.
To summarize, this remarkable article contains the first example of a physically relevant dynamical system presenting all the characteristics of chaos. Individual trajectories are unstable but their asymptotic behavior seems to be insensitive to initial conditions: they converge to the same attractor. None of the above assertions are justified, at least in the mathematical sense. How frustrating! The observed behavior happens to be robust: if one slightly perturbs the differential equation, for instance by modifying the values of the parameters, or by adding small terms, then the new differential equation will feature the same type of attractor with the general aspect of a branched surface.
This property would be rigorously established much later by Guckhenheimer and Williams.
The mathematical activity in dynamical systems during this period followed an independent and parallel path, under the lead of Smale. How can one understand this lack of communication between Lorenz—the MIT meteorologist—and Smale—the Berkeley mathematician? No bridge between different sciences was available. Mathematicians had no access to the Journal of Atmospheric Sciences.
The main tool was Morse theory describing the gradient of a generic function.
The dynamics of such a gradient is far from chaotic: trajectories go uphill and converge to some equilibrium point. Smale initiated a grandiose program aiming at a qualitative description of the trajectories of a generic vector field on compact manifolds. He conjectured that a generic vector field has a finite number of equilibrium points, a finite number of periodic trajectories, and that every trajectory converges in the future and in the past towards an equilibrium or a periodic trajectory.
He was therefore proposing that chaos does not exist!
3-in-1 Set – Concept Map Books- Math, Physics, Chemistry for Class XI & XII | IIT, JEE, NEET
Looking back at this period, Smale wrote a , b : It is astounding how important scientific ideas can get lost, even when they are aired by leading scientific mathematicians of the preceding decades. Smale realized soon by himself 4 that the dynamics of a generic vector field is likely to be much more complicated than he had expected.
He constructed a counterexample to his own conjecture Smale The famous horseshoe is a simple example of a dynamical system admitting an infinite number of periodic trajectories in a stable way. For obvious reasons, this map is called the first return map. Clearly the description of the dynamics of X reduces to the description of the iterates of F. Conversely, in many cases, one can construct a vector field from a map F.
It is often easier to draw pictures in D since it is one dimension lower than B. More importantly, Smale shows that his example is structurally stable.
Andronov and Pontryagin had introduced this concept in but in a very simple context, certainly not in the presence of an infinite number of periodic trajectories.
The proof that the horseshoe map defines a structurally stable vector field is rather elementary. The article is written in a lively, witty, and often jocular style and is full of captivating observations. Afterwards the theory progressed at a fast pace.
Smale quickly generalized the horseshoe; see for instance Smale Anosov proved in that the geodesic flow on a manifold of negative curvature is structurally stable Anosov 5. For this purpose, he created the concept of what is known today as Anosov flows.This property would be rigorously established much later by Guckhenheimer and Williams.
Individual trajectories are unstable but their asymptotic behavior seems to be insensitive to initial conditions: they converge to the same attractor. Limits of Concern have been introduced into the ERA requirements for GMOs to delimit those observed adverse effects which are not likely to cause environmental harm from those effects which have the potential to cause harm with respect to an identified protection goal in the agro-environment [ 6 ].
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