User:Salix alba/Cycle studies

From Wikipedia, the free encyclopedia

Cycles are series of states or conditions that repeat themselves, usually after a regular or nearly regular period. Cyclic behaviour is one kind — the simplest, one could say — of oscillation. The standard mathematical model of a cycle is the periodic function. Mathematicians study both periodic functions and almost periodic functions.

Cycles may be due to restorative forces causing repetition as in simple harmonic motion, regularity of motion such as daily, monthly, yearly, and other astronomical cycles, or being affected by something else that has these qualities. These forces may be physical, biological, economic or social.

The study of cycles have been a recurring phenomena throughout human history. In some cases these cycles are real physical phenomena but in other cases the cycles are illusions and represent man's fondness for seeing patterns in everything.

1 History
2 Astronomy
3 Social sciences
4 See also
5 References
6 External links
- 6.1 Groups Doing Research
- 6.2 Books & Materials On Cycle Theory
  - 6.2.1 Related Material and Links:
7 Some modern mathematical theorems involving cycles
8 Dewey links

[edit] History

Early studies of cycles are found in Vedic, Buddhist and Christian sacred books. The Vedic Timeline consists of a complex pattern of cycles with cycles ranging from 4million years to 311 trillion years. Pythagoras' study of music and Ptolemy's motion of the planets were early scientific studies of cycles; see also interval cycle and song cycle. Early studies of cycles were generally related to astronomical, astrological, and weather/climatic cycles. Copernicus, Tycho Brahe, Kepler, Newton and Einstein contributed to ever refined understanding of the motion of the planets.

Piccardi discovered inexplicable fluctuations in the rate of chemical reactions and Takata found that a human blood test he devised varied with time and depended on the sunspot cycle. Later Simon Shnoll and other researchers at the Russian Academy of Sciences found that such cycles were simultaneously happening in the growth of single celled organisms and in radioactive decay.

General cycles research pioneers were Chizhevsky in Russia and Raymond Wheeler in the USA. Edward R Dewey, who formed the Foundation for the Study of Cycles in 1942, stated that everything that had been studied had been found to have cycles present. This includes cosmology, physics, biology, geology, climate, economics, sociology, civilisation, and history in general. It was Dewey who formulated the concepts of common cycle periods, cycle synchrony and harmonically related cycle periods.

Through the middle part of the 20th century Abbott claimed to measure variations in the solar constant and links between the sunspot cycle and weather cycles on earth were vehemently denied by many scientists. With the coming of the artificial satellite age we have vastly better measures of both solar output and weather around the world and these cyclic links are now firmly established and accepted.

Edward R Dewey discovered that different disciplines often reported similar cycle periods and the phases were very similar, a principle that he called cycle synchrony. He also found that these commonly reported cycle periods were often related by ratios of 2 and 3 but was unable to determine why. An explanation for this has been offered by Ray Tomes in the harmonics theory.

The Foundation for the Study of Cycles in the US and CIFA in Europe and Russia are organisations set up to study cycles and fluctuating phenomena. An interdisciplinary Internet discussion group on cycles allows cycles researchers from different fields to exchange ideas and results.

[edit] Astronomy

Although weather changes from one summer to the next and from one winter to the next, the astronomical cycles that cause seasonal changes may to a fairly good approximation undergo identical repetitions. But astronomers also concern themselves with cycles like the 11-year sunspot cycle, in which the length of one cycle may differ from the next by an amount that cannot be neglected even over a short run of just a few cycles.

Our understanding of weather cycles being attributable to annual and monthly astronomical motions was extended by Milutin Milankovitch who showed that ice ages were closely related to variations in the Earth's orbital eccentricity, axial tilt and the precession of the equinoxes which have periods of around 100,000 years, 40,000 years and 26,000 years respectively.

The Metonic cycle in astronomy is a common multiple of the tropical year and the synodic month, and was used in several calendar systems.

The full moon cycle is an important cycle for earth, it governs the rise and fall of tides and the spring and neep tides are strongly correlated with the phases of the moon. The Menstrual cycle is said by many to be linked with the full moon cycle.

Astronomical cycles have been linked to various biological phenomena. Lyall Watson in his contreversal 1973 book Supernature discusses many strange phemonena in nature which are apparently linked to astronomical cycles. These included 24 hour circadian rhythms observed in plants, animals, fungi and cyanobacteria. In mammals this is driven by the suprachiasmatic nucleus. Luner cycles in oysters and grunion, (a small fish).^[1]

Some theories of stonehenge are linked to long term luner cycles an the pattern of lunar precession with periods of 18.6 and 56 years.^[2]

[edit] Social sciences

In modern times economic (business) cycles were studied by Joseph Kitchen, Clement Juglar, Simon Kuznets and Nikolai Kondratieff, each of whom has an economic cycle named after them.

The study of econnomic cycles has been robustly cricicised as Pseudoscience. Philip Ball in his book Critical Mass: how one thing leeds to another devotes a chapter to Rythems of the market place and examining the history of the study.

The truth is that dips and peaks in the economy resolutely refuse to recur in any predictiable manner, making attempts to construct cyclic theories of economics look increasing like Ptolemy's elaborate scheme for predicting the motions of the planets.

Ball then discusses more modern theories which investigate the precence of chaos in the buisness cycle. This work shows that economic cycles are neither completly periodic or completly random. In 1995 physicists R. Mantegna and G. Stanley analyzed over a million records of stock market indices from the previous five years, they found that the actual distribution lay between the Gaussian random walks and Lévy flights.^[3] ^[4]

In historiography and sociology, there is a theory that human history is repeating itself: the social cycle theory.

In social movement studies there is the concept of a protest cycle (or a cycle of contention). Waves of democracy concept can also be viewed as 'cycles of democratization'.

[edit] See also

[edit] References

^ Lyall Watson, Supernature, Corgie, 1973. ISBN 0340188332
^ Gerald S. Hawkins, Stonehenge Decoded Fontana, 1965.
^ Philip Ball, Critical Mass Random House 2004. ISBN 00994557865.
^ Rosario N. Mantegna, H. Eugene Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press (Cambridge, 1999)

[edit] External links

Foundation for the Study of Cycles, Inc. Visit the original foundation as established by Edward Dewey in 1941 for interdisciplinary international cycles studies
Cycles Research Institute for interdisciplinary cycles studies
Dictionary of the History of Ideas: Cosmic Cycles and cycles in human affairs
Cycles of Time Explores how the 360-day year when adjusted to a true solar year creates increasingly large cycles of time.

Is the study and interpretation of rhythmic phenomena. Since the separation of religion and science, people has been trying to discover, understand, and explain the nature and origin of cycles. Much emphasis has been placed in modern times with unravelling the mystery of recurrent cycles which are observed in the market economies. These rhythmic phenomena been observed in both the natural and the social sciences as well. A huge driving force behind this theory is the market economy in particular stock market and the commodities markets. Hedge funds and other instituation have spent millions in research and deploy mathematicians to develop mathematical models to help predict market cycles in order the hedge their investments. Other mathamatician and have sought the Holy Gail equation(s) that would unlock the cyclical patterns of the finical markets. This idea was explored in the hit indi film Pi.

[edit] Groups Doing Research

Private

Cycles Research Institute -- is to discover the causes and conditions for already observed and cataloged cyclic and rhythmic behaviors; Then, to classify the causes and conditions within a sciences method. And finally, to incorporate these causes and conditions into the mainstream of modern scientific theory and knowledge.
The Ludwig von Mises Institute 1

Public

CEPA School. overview of studies-- surveys of Old Business Cycle Theory and Keynesian Business Cycle Theory.
Harvard University (business Cycle Theory: 1 2)
Auburn University

[edit] Books & Materials On Cycle Theory

Business

Hayekian Trade Cycle Theory: A Reappraisal 1: Auburn University.
A critique of structural VARs using business cycle theory [1]
The Hedge Fund Edge : Maximum Profit/Minimum Risk Global Trend Trading Strategies - by Wiley Trading
Economic Alchemy and Liquidity Cycle Theory
K-Cycle Theory (1)
Business Cycle Theory ISBN-10: 0-19-925682-9 and ISBN-13: 978-0-19-925682-2 1

Life Sciences

Geographical Cycle Theory1 See also Gaia Theory
Power Cycle Theory and Global Politics (reference series series
Social cycle theory

[edit] Related Material and Links:

Modern Business Cycle Theory Harvard University Press: Modern Business Cycle Theory -- 1
Business Cycle
Keynesian economics
Friedrich August von Hayek - monitary cycle theory

[edit] Some modern mathematical theorems involving cycles

While the table above (like most of Dewey's work) amounts to numerology, we can still ask: why are cycles so common in many-- but certainly not all-- phenomena in subjects such as economics, biology, and physics?

It goes beyond the scope of this article to attempt to answer this question, but we can still ask: what does modern mathematics have to say concerning cyclic behavior common to many phenomena in economics, biology, and physics?

As it turns out, there are several distinct (but related) notions of cycle which are studied in various areas of modern mathematics, particularly the modern theory of dynamical systems. Even better, there are several important and very striking theorems in which cycles play a leading role. Some of the better known examples include:

Sarkovskii's theorem reveals the unexpected role played by multiples of two and three in answering the question: what periods occur in a dynamical system obtained by iterating a continuous map on the unit interval? The Sharkovskii theorem gives a complete answer to this question and thereby provides deep insight into these systems (which are more general than they might appear). In particular, it implies that if such a system has a three-cycle, it has n-cycles for every positive integer n!
Hurwitz's theorem on the hierarchy of irrationality answers the question: which quadratic irrationals are most difficult to accurately approximate by rational numbers in lowest terms which have small denominators? This might at first appear unrelated to cyclic phenomena, but the earliest important theorems in Greek mathematics involve rational approximations, in the context of devising accurate but practical calendars, modeling astronomical motion, and so forth. Hurwitz's answer, by the way, is: the Golden mean (plus some closely related numbers).
The KAM theorem provides a satisfying answer to the question: when we perturb a Hamiltonian system, why do certain invariant tori persist longer than others? Here, each invariant torus is associated with quasiperiodic orbits. It turns out that the golden mean is associated with the most resistant tori, which gives a connection with Hurwitz's theorem! Applications of the KAM theorem include the orbits of solar system objects in complicated multi-body scenarios.
In symbolic dynamics, Bowen's formula gives the Artin-Mazur zeta function of a one-dimensional shift of finite type as a rational function (the reciprocal of a polynomial, in fact) which is easily computed from the graph defining the shift. Knowledge of the zeta function (and Möbius inversion) enables one to efficiently determine which periods occur and how many points have a given period. This formula is relatively easy, but it leads to a deep connection with the dynamics of closed geodesics on compact hyperbolic surfaces, and even an intriguing connection with the prime number theorem! See prime geodesic.

The situation can be summed up like this:

many famous mathematicians have studied many different notions of cycles,
the results of their work includes several of the most memorable theorems in modern mathematics,
appropriate (mathematically defined) notions of cycles play important roles in various parts of the modern theory of dynamical systems, and therefore in the many applications of this theory to economics, biology, physics, and other areas.

It is important to understand that none of this work has anything relation with the semi-mystical speculations of Dewey. It is also important to understand that in symbolic dynamics one can exhibit dynamical systems which have no periodic points at all! Since these include systems which can be used to model real world phenomena, this fact directly contradicts the alleged claim by Dewey that all phenomena include cyclic behavior.

Nonetheless, members of the Foundation for the Study of Cycles have attempted to use mainstream research resting upon some sophisticated mathematical theories (including the theorems outlined above), in order to argue that the very existence of this work demonstrates (they allege) that Dewey's speculations constitute accepted science. This claim is not supported by examination of the mainstream research literature.