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Can the flap of a butterfly’s wings in Brazil set off a tornado in Texas? This intriguing hypothetical scenario, commonly called “the butterfly effect,” has come to embody the popular conception of a chaotic system, in which a small difference in initial conditions will cascade toward a vastly different outcome in the future.

Understanding and modeling chaos can help address a variety of scientific and engineering questions, and so researchers have worked to develop better mathematical definitions of chaos. These definitions, in turn, will aid the construction of models that more accurately represent real-world chaotic systems.

Now, researchers from the University of Maryland have described a new definition of chaos that applies more broadly than previous definitions. This new definition is compact, can be easily approximated by numerical methods and works for a wide variety of chaotic systems. The discovery could one day help advance computer modeling across a wide variety of disciplines, from medicine to meteorology and beyond. The researchers present their new definition in the July 28, 2015 issue of the journal Chaos.

“Our definition of chaos identifies chaotic behavior even when it lurks in the dark corners of a model,” said Brian Hunt, a professor of mathematics with a joint appointment in the Institute for Physical Science and Technology (IPST) at UMD. Hunt co-authored the paper with Edward Ott, a Distinguished University Professor of Physics and Electrical and Computer Engineering with a joint appointment in the Institute for Research in Electronics and Applied Physics (IREAP) at UMD.

The study of chaos is relatively young. MIT meteorologist Edward Lorenz, whose work gave rise to the term “the butterfly effect,” first noticed chaotic characteristics in weather models in the mid-20th century. 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.

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

Sourced through Scoop.it from: phys.org

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