Forum for Science, Industry and Business

Sponsored by:     3M 
Search our Site:

 

A New Litmus Test for Chaos?

29.07.2015

Researchers hope the concept of "expansion entropy" will become a simple, go-to tool to identify (sometimes hidden) chaos in a wide range of model systems -- and could help distinguish which chaotic systems could be subject to some measure of control

Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas? This intriguing question -- the title of a talk given by MIT meteorologist Edward Lorenz at a 1972 meeting -- has come to embody the popular conception of a chaotic system, one in which a small difference in initial conditions will cascade toward a vastly different outcome in the future.


Images courtesy of the Chaos Group at the University of Maryland.

Researchers from the University of Maryland have proposed a new definition of chaos that applies to a wide variety of chaotic systems, including attractors, illustrated in the top left corner by a picture of a Tinkerbell attractor, repellers, as illustrated by the top and bottom right side pictures of chaotic scattering, and forced systems, as illustrated by the lower left picture of the motions of a forced damped pendulum. The lower right picture shows the pattern of light created inside a pyramid of 4 reflecting balls, work that was done by D.Sweet, E.Ott, J.Yorke, D.Lathrop and B.Zeff at the University of Maryland.

Mathematically, extreme sensitivity to initial conditions can be represented by a quantity called a Lyapunov exponent, which is positive if two infinitesimally close starting points diverge exponentially as time progresses. Yet Lyapunov exponents as a definition of chaos have limitations -- they only test for chaos in particular solutions of a model, not in the model itself, and they can be positive, for example in simple scenarios of unlimited growth, even when the underlying model is considered too straightforward to be deemed chaotic.

Now researchers from the University of Maryland have come up with a new definition of chaos that applies more broadly than Lyapunov exponents and other previous definitions of chaos. The new definition fits on a few lines, can be easily approximated by numerical methods, and works for a wide variety of chaotic systems. The researchers present the definition in a paper in the 25th anniversary issue of the journal Chaos, from AIP Publishing.

Hunting Down Hidden Chaos

Edward Lorenz, the scientist whose work gave rise to the term "the butterfly effect," first noticed chaotic characteristics in weather models. In 1963 he published a set of differential equations to describe atmospheric airflow and noted that tiny variations in initial conditions could drastically alter the solution to the equations over time, making it difficult to predict the weather in the long-term.

The chaotic solution to Lorenz's equations looks, fittingly, like two wings of a butterfly. The shape can be categorized, in math-speak, as an attractor, meaning it is easy to identify with Lyapunov exponents, said Brian Hunt, a mathematician at the University of Maryland and a member of the university's "Chaos Group." Yet not all chaotic behavior is quite so clear, he said.

As an example Hunt describes four Christmas balls stacked in a pyramid, a set-up analyzed by Hunt's colleagues David Sweet, Edward Ott, James Yorke, and others at the University of Maryland. Light hitting the shiny spheres reflects off in all directions. Most of the light travels simple paths, but occasional photons can become trapped in the interior of the pyramid, bouncing chaotically back and forth off the ornaments. The chaotic, one-off light paths are mathematically categorized as repellers, and can be difficult to find from model equations unless you know exactly where to look.

"Our definition of chaos identifies chaotic behavior even when it lurks in the dark corners of a model," said Hunt, who collaborated on the paper with Edward Ott, a professor at the University of Maryland and the author of the graduate textbook "Chaos in Dynamical Systems." The two researchers also broadened the definition by including systems that are forced, meaning that external factors continue to push or pull on the model as it evolves.

Researchers commonly encounter chaotic repellers, found in physical systems such as water flowing through a pipe, asteroid orbits, and chemical reactions, and forced systems, found for example in bird flocks, geophysics and the way the body controls the heartbeat.

Calculating Uncertainty

To fit the generally recognized forms of chaos under one umbrella definition, Hunt and Ott turned to a concept called entropy. In a system that changes over time, entropy represents the rate at which disorder and uncertainty build up.

The idea that entropy could be a proxy for chaos is not new, but the standard definitions of entropy, such as metric entropy and topological entropy, are trapped in the mathematical equivalent of a straightjacket. The definitions are difficult to apply computationally, and have stringent prerequisites that disqualify many physical and biological systems of interest to scientists.

Hunt and Ott defined a new type of flexible entropy, called expansion entropy, which can be applied to more realistic models of the world. The definition can be approximated accurately by a computer and can accommodate systems, like regional weather models, that are forced by potentially chaotic inputs. The researchers define chaotic models as ones that exhibit positive expansion entropy.

The researchers hope expansion entropy will become a simple, go-to tool to identify chaos in a wide range of model systems. Pinpointing chaos in a system can be a first step to determining whether the system can ultimately be controlled.

For example, Hunt explains, two identical chaotic systems with different initial conditions may evolve completely differently, but if the systems are forced by external inputs they may start to synchronize. By applying the expansion entropy definition of chaos and characterizing whether the original systems respond chaotically to inputs, researchers can tell whether they can wrestle some control over the chaos through inputs to the system. Secure communications systems and pacemakers for the heart would be just two of the potential applications of this type of control, Hunt said.

The article, "Defining Chaos," is authored by Brian R. Hunt and Edward Ott. It will be published in the journal Chaos: An Interdisciplinary Journal of Nonlinear Science on July 28, 2015 (DOI: 10.1063/1.4922973). After that date, it can be accessed at: http://scitation.aip.org/content/aip/journal/chaos/25/9/10.1063/1.4922973 
The authors of this paper are affiliated with the University of Maryland.

ABOUT THE JOURNAL

Chaos: An Interdisciplinary Journal of Nonlinear Science is devoted to increasing the understanding of nonlinear phenomena and describing the manifestations in a manner comprehensible to researchers from a broad spectrum of disciplines. See: http://chaos.aip.org/

Contact Information
Jason Socrates Bardi
American Institute of Physics
jbardi@aip.org

240-535-4954

@jasonbardi

Jason Socrates Bardi | newswise

More articles from Physics and Astronomy:

nachricht Extremely fine measurements of motion in orbiting supermassive black holes
28.06.2017 | Stanford University

nachricht Ultra-compact phase modulators based on graphene plasmons
27.06.2017 | ICFO-The Institute of Photonic Sciences

All articles from Physics and Astronomy >>>

The most recent press releases about innovation >>>

Die letzten 5 Focus-News des innovations-reports im Überblick:

Im Focus: Can we see monkeys from space? Emerging technologies to map biodiversity

An international team of scientists has proposed a new multi-disciplinary approach in which an array of new technologies will allow us to map biodiversity and the risks that wildlife is facing at the scale of whole landscapes. The findings are published in Nature Ecology and Evolution. This international research is led by the Kunming Institute of Zoology from China, University of East Anglia, University of Leicester and the Leibniz Institute for Zoo and Wildlife Research.

Using a combination of satellite and ground data, the team proposes that it is now possible to map biodiversity with an accuracy that has not been previously...

Im Focus: Climate satellite: Tracking methane with robust laser technology

Heatwaves in the Arctic, longer periods of vegetation in Europe, severe floods in West Africa – starting in 2021, scientists want to explore the emissions of the greenhouse gas methane with the German-French satellite MERLIN. This is made possible by a new robust laser system of the Fraunhofer Institute for Laser Technology ILT in Aachen, which achieves unprecedented measurement accuracy.

Methane is primarily the result of the decomposition of organic matter. The gas has a 25 times greater warming potential than carbon dioxide, but is not as...

Im Focus: How protons move through a fuel cell

Hydrogen is regarded as the energy source of the future: It is produced with solar power and can be used to generate heat and electricity in fuel cells. Empa researchers have now succeeded in decoding the movement of hydrogen ions in crystals – a key step towards more efficient energy conversion in the hydrogen industry of tomorrow.

As charge carriers, electrons and ions play the leading role in electrochemical energy storage devices and converters such as batteries and fuel cells. Proton...

Im Focus: A unique data centre for cosmological simulations

Scientists from the Excellence Cluster Universe at the Ludwig-Maximilians-Universität Munich have establised "Cosmowebportal", a unique data centre for cosmological simulations located at the Leibniz Supercomputing Centre (LRZ) of the Bavarian Academy of Sciences. The complete results of a series of large hydrodynamical cosmological simulations are available, with data volumes typically exceeding several hundred terabytes. Scientists worldwide can interactively explore these complex simulations via a web interface and directly access the results.

With current telescopes, scientists can observe our Universe’s galaxies and galaxy clusters and their distribution along an invisible cosmic web. From the...

Im Focus: Scientists develop molecular thermometer for contactless measurement using infrared light

Temperature measurements possible even on the smallest scale / Molecular ruby for use in material sciences, biology, and medicine

Chemists at Johannes Gutenberg University Mainz (JGU) in cooperation with researchers of the German Federal Institute for Materials Research and Testing (BAM)...

All Focus news of the innovation-report >>>

Anzeige

Anzeige

Event News

Plants are networkers

19.06.2017 | Event News

Digital Survival Training for Executives

13.06.2017 | Event News

Global Learning Council Summit 2017

13.06.2017 | Event News

 
Latest News

Extensive Funding for Research on Chromatin, Adrenal Gland, and Cancer Therapy

28.06.2017 | Awards Funding

Predicting eruptions using satellites and math

28.06.2017 | Earth Sciences

Extremely fine measurements of motion in orbiting supermassive black holes

28.06.2017 | Physics and Astronomy

VideoLinks
B2B-VideoLinks
More VideoLinks >>>