Late in his career, Lorenz began to be recognized with international accolades for the importance of his work on deterministic chaos. Lorenz is remembered by colleagues and friends for his quiet demeanor, gentle humility, and love of nature. In , The Lorenz Center, a climate think tank devoted to fundamental scientific inquiry, was founded at MIT in honor of Lorenz and his pioneering work on chaos theory and climate science. A video produced for the event highlights the indelible mark made by Charney and Lorenz on MIT and the field of meteorology as a whole.

Lorenz published many books and articles, a selection of which can be found below. A more complete list can be found on the The Lorenz Center website: link. From Wikipedia, the free encyclopedia. West Hartford, Connecticut , United States. Cambridge, Massachusetts , United States.

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Systems science portal. Biographical Memoirs of Fellows of the Royal Society. Physics Today. Bibcode : PhT Chaos at fifty , Physics Today 66 5 , The New York Times. Retrieved MIT News Office. Archived from the original on September 23, Washington, D.

## Lorenz and the Essence of Chaos

A method of applying the hydrodynamic and thermodynamic equations to atmospheric model Massachusetts Institute of Technology. Massachusetts Institute of Technology. MIT Tech Talk. World Meteorological Organization. The Lorenz Center. Cambridge, MA. Journal of the Atmospheric Sciences. Bibcode : JAtS Chaos: Making a New Science.

London: Cardinal. Lorenz Archived from the original on Bulletin of the American Meteorological Society. Bibcode : BAMS MIT Technology Review. Systems science. Doubling time Leverage points Limiting factor Negative feedback Positive feedback. Alexander Bogdanov Russell L. Hall Lydia Kavraki James J. Kay Faina M. Systems theory in anthropology Systems theory in archaeology Systems theory in political science.

List Principia Cybernetica. Category Portal Commons. Chaos theory. Yorke Lai-Sang Young. Fellows of the Royal Society elected in When the times are those of consecutive strikes on a pin, the expressions will amount to nothing more than a system of difference equations, which in this case will have been derived by solving the differential equations. Thus a mapping will have been derived from a flow. Indeed, we can create a mapping from any flow simply by observing the flow only at selected times.

If there are no special events, like strikes on a pin, we can select the times as we wish—for instance, every hour on the hour. Very often, when the flow is defined by a set of differential equations, we lack a suitable means for solving them—some differential equations are intrinsically unsolvable.

## An introduction to Chaos Theory | PSpice

In this event, even though the difference equations of the associated mapping must exist as relationships, we cannot find out what they look like. For some real-world systems we even lack the knowledge needed to formulate the differential equations; can we honestly expect to write any equations that realistically describe surging waves, with all their bubbles and spray, being driven by a gusty wind against a rocky shore? If the pinball game is to chaos what the coin toss is to complete randomness, it has certainly not gained the popularity as a symbol for chaos that the coin has enjoyed as a symbol for randomness.

The butterfly. I noted also that a single flap would have no more effect on the weather than any flap of any other The paper is reproduced in its original form as Appendix 1. The thing that has made the origin of the phrase a bit uncertain is a peculiarity of the first chaotic system that I studied in detail. A number of people with whom I have talked have assumed that the butterfly effect was named after this attractor.

Perhaps it was. Here the death of a prehistoric butterfly, and its consequent failure to reproduce, change the outcome of a present-day presidential election. Before the Washington meeting I had sometimes used a sea gull as a symbol for sensitive dependence. The switch to a butterfly was actually made by the session convenor, the meteorologist Philip Merilees, who was unable to check with me when he had to submit the program titles. Perhaps the butterfly, with its seeming frailty and lack of power, is a natural choice for a symbol of the small that can produce the great.

## The essence of chaos

Other symbols have preceded the sea gull. In George R. The heart will proceed to beat at irregular intervals and sometimes with varying intensity, instead of ticking like a metronome. It has been conjectured that arrhythmias are manifestations of chaos. Clearly they entail an absence of some Precise definitions are not always convenient ones.

Having defined chaos in terms of sensitive dependence, we may discover that it is difficult to determine whether certain phenomena are chaotic.

### More By No Alternative

The pinball machine should present no problem. If instead our system is a flag flapping in the wind, we do not have this option. Perhaps on two occasions we might hold the flag taut with some stout string, and let the initial moments be the times when the strings are cut. If we then worry that subsequent differences in behavior may result from fluctuations in the wind instead of from intrinsic properties of the flag, we can circumvent the problem by substituting an electric fan. Nevertheless, we shall have introduced highly unusual initial states—flags in a breeze do not ordinarily become taut—and these states may proceed to evolve in a highly unusual manner, thereby invalidating our experiment as a test for chaos.

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Fortunately there is a simpler approach to systems like the flapping flag, and it involves looking for rhythm. Before we can justify it, we must examine a special property of certain dynamical systems, which is known as compactness. Suppose that on a round of golf you reach the tee of your favorite par-three hole and drive your ball onto the green.

If you should decide to drive a second ball, can you make it come to rest within a foot of the first one, if this is what you wish?

Presumably not; even without any wind the needed muscular control is too great, and the balls might not be equally resilient. If instead you have several buckets of balls and continue to drive, you will eventually place a ball within a foot of one that you have already driven, although perhaps not close to the first one. This will not be because The argument still holds if the critical distance between the centers of two balls is one centimeter, or something still smaller, as long as your caddy removes each ball and marks the spot before the next drive.

Of course you may need many more buckets of balls. The surface of the green is two-dimensional—a point on it may be specified by its distance and direction from the cup—and its area is bounded. Many dynamical systems are like the ball on the green, in that their states may be specified by the values of a finite number of quantities, each of which varies within strict bounds. If in observing one of these systems we wait long enough, we shall eventually see a state that nearly duplicates an earlier one, simply because the number of possible states, no one of which closely resembles any other one, is limited.

Systems in which arbitrarily close repetitions—closer than any prespecified degree—must eventually occur are called compact. For practical purposes the flag is a compact system.

The bends in the flag as it flaps often resemble smooth waves. Let us define the state of the flag by the position and velocity of each point of a wellchosen grid, perhaps including the centers of the stars if it is an American flag, instead of using every point on the flag. Two states that, by this definition, are nearly alike must then eventually occur, and any reasonable interpolation will indicate that the positions and velocities of any other points on the flag will also be nearly alike in the two instances.

Our pinball machine is not a compact system. Not only do near repetitions not have to occur while a single ball is in play, but they cannot, since friction is continually tending to slow the ball, and the only way that the ball can regain or maintain its speed is to roll closer to the base of the machine.

However, we can easily visualize a modified system where near repetitions are inevitable. Imagine a very long pinball machine; it might stand on a gently sloping sidewalk outside a local drugstore, and extend for a city block. Let the playing surface be marked off into sections, say one meter long, and let the arrangement of the pins in each section be identical with that in any other.

Except for being displaced from each other by one or more sections, the complete paths of two balls, occupying similar positions in different sections, and moving with These quantities all vary within limited ranges. It follows that if the city block is long enough, a near repetition of a previous state must eventually occur. In Figure 3 we see the computed path of the center of a single ball as it works its way past eighty pins, in a machine that is not only elongated but very narrow; the playing space is only twice as wide as the ball.

Each pin is set one-quarter of the way in from one side wall or the other. The long machine is displayed as four columns; the right-hand column contains the first twenty pins, and each remaining column is an extension of the one to its right.

The vertical scale has been compressed, as if we were looking from a distance with our eye just above the playing surface, so that the circular upper end appears elliptical, as would an area directly below the ball, if it were shown. You may find the figure easier to study if you give it a quarter turn counterclockwise and look at the path as a graph. Evidently the ball completely misses about a third of the pins, but the rebounds from pins somewhat outnumber those from side walls. We see that the path from above the first to below the seventh pin in the third column nearly duplicates the path in the same part of the second column.