Chaos

Overview

Course No. 1333

Explore a revolutionary field of mathematics and discover new ways to think about the world with this mind-expanding course. It has been called the third great revolution of 20th-century physics, after relativity and quantum theory. But how can something called chaos theory help you understand an orderly world? What practical things might it be good for? What, in fact, is chaos theory?

“Chaos theory,” according to Dr. Steven Strogatz, Director of the Center for Applied Mathematics at Cornell University, “is the science of how things change.” It describes the behavior of any system whose state evolves over time and whose behavior is sensitive to small changes in its initial conditions. Surprisingly, you have already encountered chaos theory before, although you might not have recognized it at the time. From the flapping of a butterfly’s wings to the dripping of a leaky faucet, chaos theory draws a wealth of unordinary insight from the most ordinary of occurrences.

Chaos theory affects nearly every field of human knowledge and endeavor, from astronomy and zoology to the arts, the humanities, and business. It can help analysts understand price fluctuations in the stock market, ensure a smooth flow of data traffic on the Internet, and show insurance companies how to manage the risks of natural catastrophes. Whether charting the exciting history of the field, focusing on fractals as “the footprints of chaos,” or journeying to the frontiers of chaos research, this course shows you new ways to think about and view the world around you.

Video Lectures

01: The Chaos Revolution

Chaos was once ignored by traditional science but is now both a pop sensation and a tremendously important field. But what is the science of chaos and why is it revolutionary and important?

Duration: 32 min

02: The Clockwork Universe

The scientific revolution launched by Galileo, Kepler, and Newton left a great legacy: the idea of an orderly universe ruled by mathematical laws. But is there something disquieting in the idea of a vast, impersonal, clockwork universe of determinism with no room for chance?

Duration: 29 min

03: From Clockwork to Chaos

By the late 19th century, three cracks appeared in determinism’s foundations: relativity, quantum mechanics—and chaos. The “three-body problem” was considered THE mathematical challenge of the era, and its solution, involving a still-unimagined chaos, eluded some of mathematics’ greatest minds.

Duration: 31 min

04: Chaos Found and Lost Again

Henri Poincaré’s groundbreaking work on the three-body problem implied that a system governed by deterministic laws could still be unpredictable; chaos had crept into the clockwork. Although Poincaré invented a new, “visual” way of thinking about the mathematics involved, his brilliant discovery was quickly forgotten.

Duration: 30 min

05: The Return of Chaos

For 70 years, chaos remained a scientific backwater. The calm ended with a thunderclap from a man fascinated by storms and weather. You see how Edward Lorenz discovered chaos in a model of weather patterns that allowed him to happen upon the “butterfly effect.”

Duration: 30 min

06: Chaos as Disorder-The Butterfly Effect

The butterfly effect—the extreme sensitivity of a chaotic system to tiny changes in its initial conditions—has become part of popular culture but is frequently misunderstood. You begin to understand not only its importance and power but also its limitations.

Duration: 32 min

07: Picturing Chaos as Order-Strange Attractors

Your introduction to chaos has highlighted its unpredictable, random side, as exemplified by the butterfly effect. But there is also an amazing order inherent in chaos, and you learn how this can be visualized through the infinitely complex image known as a “strange attractor.”

Duration: 29 min

08: Animating Chaos as Order-Iterated Maps

If a strange attractor is analogous to an image created through time-lapse photography, Lorenz’s “iterated map” might be the product of a series of strobe-light photographs. But despite its profound implications, Lorenz’s discovery failed to attract the scientific community’s notice.

Duration: 32 min

09: How Systems Turn Chaotic

By the 1970s, there was an unprecedented convergence of disciplines. Researchers in mathematics, ecology, and fluid mechanics found themselves asking the same question: How does an orderly system suddenly turn chaotic? You see how a famous iterated map known as the logistic map reveals the most basic route.

Duration: 32 min

10: Displaying How Systems Turn Chaotic

You deepen your understanding of the logistic map with the icon of chaos known as the orbit diagram. Its breathtaking imagery amounts to a Rosetta Stone for making sense of certain forms of chaos in the natural world.

Duration: 31 min

11: Universal Features of the Route to Chaos

In 1978, physicist Mitchell Feigenbaum made a stunning breakthrough, showing that the logistic map displayed universal features so generic that they must also occur in nature, even though no laws of nature are built into it. You begin to understand how such universality arises.

Duration: 30 min

12: Experimental Tests of the New Theory

In the early 1980s, painstaking experiments on such disparate systems as swirling fluids, electronic circuits, and oscillating chemical reactions confirmed the predictions of chaos theory. Overreaching by some advocates, however, has provoked a backlash of skepticism to this day.

Duration: 30 min

13: Fractals-The Geometry of Chaos

The pioneers of chaos were bewildered by the fantastic shapes they encountered while trying to visualize chaos. In the first of several lectures devoted to these intricate shapes—now called fractals—you learn why they are so inextricably connected to chaos.

Duration: 32 min

14: The Properties of Fractals

You are introduced to the two most distinctive properties of fractals—inexhaustible structural richness and “self-similarity”—or the resemblance of the parts to the whole—before learning how the science of fractals came into being and its situation in the broader scientific landscape.

Duration: 31 min

15: A New Concept of Dimension

Using some idealized geometric examples, you learn how to define the dimension of a fractal—discovering that the usual categories of one-, two-, or three-dimensional usually do not apply, and that fractals are so convoluted they fall somewhere in between, such as 1.26-dimensional!

Duration: 31 min

16: Fractals Around Us

Fractals are not merely static geometric shapes but also can represent erratic processes “in time,” such as fluctuating stock prices, Internet data bursts, or earthquakes. You learn that their gyrations are wilder and more frequent than conventional statistical methods would predict and make their management more complex.

Duration: 31 min

17: Fractals Inside Us

From lungs to nervous systems to the nutrient supply systems of plants, all living things are built from fractal networks. You examine this geometry of life, including a recent theory that invokes fractal architecture to explain one of the most comprehensive laws in biological science.

Duration: 32 min

18: Fractal Art

This lecture shows you some of the manifestations of fractals in art, including the controversial drip paintings of Jackson Pollock. Some have suggested that they contain fractal characteristics that changed over the course of his career in a very systematic way.

Duration: 31 min

19: Embracing Chaos-From Tao to Space Travel

Does chaos have practical applications? Because tiny nudges to a chaotic system can have potent effects, these systems are exceptionally responsive. You see the advantages of harnessing chaos in the dramatic story of how a NASA mathematician “surfed” the gravitational field to salvage a Japanese lunar mission gone wrong.

Duration: 29 min

20: Cloaking Messages with Chaos

Although the feasibility of encrypting electronic messages by cloaking them in chaotic “noise” has been verified in real-world tests, questions remain. Could an eavesdropper crack the chaos? This lecture shows you what such an application could mean in a world of growing concerns about cyberterrorism, national security, and cell phone and Internet privacy.

Duration: 31 min

21: Chaos in Health and Disease

Building on decades of biological research, chaos theorists have been asking questions about the dynamics of bodily rhythms. Can the mathematics of chaos help predict an epileptic seizure? Quell or prevent cardiac arrhythmias? Perhaps most controversially, can chaos in the body ever be a sign of health rather than of sickness?

Duration: 31 min

22: Quantum Chaos

Can chaos theory coexist with quantum theory? Can it survive the descent to the strange world of the atom, where Newtonian trajectories dissolve into a haze of quantum probability waves? You see how scientists reconcile two radically different views of reality.

Duration: 31 min

23: Synchronization

Large, complex systems having many interacting parts often display a remarkable capacity for organizing themselves, with their individual parts becoming synchronized. This lecture shows you systems as diverse as pendulum clocks, fireflies, heart cells, and menstrual cycles and takes you inside the opening-day swaying of London’s Millennium Bridge.

Duration: 32 min

24: The Future of Science

You review what you’ve learned and examine the future role of chaos theory. In a world where most of the major unsolved issues facing science—including cancer, consciousness, the origin of life, and AIDS—involve fundamentally “nonlinear” systems, chaos theory can be a crucial first step toward their solution.

Duration: 31 min