Chaos Theory

INTRODUCTION

 

Since the late 1960's with a new appreciation of Poincaré's work on the three body problem, we now know that many non-periodic real world pheonomena behave in a chaotic manner.  And this is not a consequence of their complexity but rather a mathematically demonstrated proof.  And the defining characteristic of chaos is that these systems are theoretically unpredictable.   And this is a fact no future scientific discovery, no improvement in computational resources, no amount of research funding can remedy.  Amazingly, these systems were previously thought to be well described by Newton's classical laws of motion and gravitation.  Instabilities in computed predictions generally arise because of some non-linear aspect as well as an extreme sensitivity to initial conditions.

 

Another difficulty is that even for simple systems there seems to be a finite limit on any predictability. Even though an increasingly better knowledge of the initial state allows a prediction further into the future, the requirements on initial accuracy increase at a faster rate than the window of predictability improves [1].  Thus even for classical systems, there is a finite window extending both into the past and the future beyond which no predictions are theoretically possible.

 

HISTORY

 

That the evolution of a few seemingly simple differential equations in a Lorenz model was inherently unpredictable in any possible computer simulation came as a real surprise in the early 1960's. But since then most real world systems, having complex relationships much beyond the early models of Lorenz, have been shown to be chaotic [2].  In 1969, Sir James Lighthill (1924–1998) was elected Lucasian Professor of Mathematics to succeed Physics Nobel Laureate, P.A.M. Dirac. This is the chair at the University of Cambridge previously held by Sir Issac Newton. As a firm believer of Newtonian mechanics, Sir James’ statement of public apology is an enlightenment to read:

 

“Here I have to pause, and to speak once again on behalf of the broad global fraternity of practitioners of mechanics. We are all deeply conscious today that the enthusiasm of our forebears for the marvelous achievements of Newtonian mechanics led them to make generalizations in this area of predictability which, indeed, we may have generally tended to believe before 1960, but which we now recognize were false. We collectively wish to apologize for having misled the general educated public by spreading ideas about the determinism of systems satisfying Newton’s laws of motion that, after 1960, were to be proved incorrect. In this lecture, I am trying to make belated amends by explaining both the very different picture that we now discern, and the reasons for it having been uncovered so late [3].”

 

CONCLUSIONS

 

The best current thinking seems to be that even for classical systems, the argument for a clockwork universe as a strict consequence of Newtonian dynamics is no longer logically valid. Since both complexity and errors accumulate over time, perhaps exponentially, we cannot be certain of determinism even for short times, or even in principle, or even for classical systems. The claims of Chaos theory are thus that nature seems to draw a curtain on predictions of mechanical motion in a clockwork universe that is forever beyond any ability to penetrate.

 

REFERENCES

 

1.      "Deterministic Nonperiodic Flow". Lorenz, (March 1963). Journal of the Atmospheric Sciences 20 (2): 130–141

2.      “Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?“, Edward N. Lorenz, American Association for the Advancement of Science (AAAS), December 29, 1972.

3.      “Sir James Lighthill and Modern Fluid Mechanics”, by Lokenath Debnath, The University of Texas-Pan American, US, Imperial College Press: ISBN 978-1-84816-113-9: ISBN 1-84816-113-1, Singapore, page 31. Online at http://cs5594.userapi.com/u11728334/docs/25eb2e1350a5/Lokenath_Debnath_Sir_James_Lighthill_and_mode.pdf