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Chaos

Chaos

Professor Steven Strogatz Ph.D.
Cornell University
Course No.  1333
Course No.  1333
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Course Overview

About This Course

24 lectures  |  30 minutes per lecture

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.

The 24 lectures of Chaos take you to the heart of chaos theory as it is understood today. Taught by Professor Strogatz, an award-winning Ivy League professor and a scientist described by

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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.

The 24 lectures of Chaos take you to the heart of chaos theory as it is understood today. Taught by Professor Strogatz, an award-winning Ivy League professor and a scientist described by Nature magazine as "one of the most creative biomathematicians of the past few decades," Chaos introduces you to a fascinating discipline that has more to do with your everyday life than you may realize.

A Revolutionary Way of Thinking

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.

This course shows you the importance of this revolutionary field and how it has helped us come closer than ever to solving some of life's mysteries. Today, the underlying mathematics of science's major unsolved problems—including the nature of consciousness, the origin of life, and cancer—are essentially nonlinear; express any of these problems as a mathematical system and you learn that the whole may be either more or less than the sum of its parts.

In its ability to tackle bewilderingly complex problems, chaos theory has revolutionized the way we perceive the world around us. It allows scientists to reach beyond a dependency on the analytical limitations of the deterministic, "clockwork" universe that was the legacy of thinkers like Galileo, Kepler, and especially Newton.

Throughout the lectures, Professor Strogatz makes the case for why chaos theory marks such a radical departure from traditional science:

  • It asks unusual questions at the everyday scale of human life.
  • It shifts the focus off the laws of nature and onto their consequences.
  • It uses the computer not as a calculating tool but as a means of amplifying intuition.
  • It does not reduce complex problems into their separate parts but puts the parts back together to help understand the whole.
  • It is radically interdisciplinary in an era of increasingly specialized disciplines.
  • It paints a topsy-turvy picture of the world in which simple systems can show complex behavior.
  • It is a scientific field in which change came about suddenly.

Follow the Exciting Story of Chaos

As you delve into this ever-evolving field, you learn the surprising tale of how chaos theory was discovered—a story that Professor Strogatz likens to a detective novel filled with twists and turns.

First glimpsed by the French mathematician Henri Poincaré, the notion of chaos theory was lost for nearly a century before being rediscovered—almost accidentally. It was revived by a mathematically oriented meteorologist named Edward Lorenz, whose development of the butterfly effect (the extreme sensitivity of a chaotic system to tiny changes in its initial conditions) had little impact until the 1970s and 1980s, when the wave of chaos theory finally crashed onto the shores of the scientific community.

As you follow the story of chaos theory's development, you approach the core ideas of chaos in the same way the world's greatest thinkers, grounded in their historical contexts, once did. This story not only helps you understand the fundamentals of this field, but it also helps you appreciate the extraordinary intellectual feat that chaos theory represents.

Learn Chaos Theory Visually

This course offers you a unique opportunity to get an expert's instruction on the field of chaos theory and is one of the only places outside the halls of academia where you can follow along with detailed computer graphics—specifically developed for this course—as visual aids.

"For understanding these core concepts [of chaos theory], pictures turn out to be much more powerful than formulas," notes Professor Strogatz. Forgoing a heavy reliance on advanced math, he uses clear and powerful computer graphics to clarify chaos theory's core concepts.

A large portion of the course explores the intimate relationship between chaos theory and fractals: shapes or processes whose structures repeat ad infinitum such that the tiniest parts resemble the original whole. You see how fractals are unique from more commonly known shapes like circles and cubes and how they can be used to describe a variety of processes and phenomena like the jagged coastline of Norway or the drip paintings of Jackson Pollock.

Find the Unordinary in the Ordinary

Professor Strogatz's expert guidance lays bare the complexities of chaos theory in a way that any interested layperson can understand. With the insights he provides in Chaos, news stories about key scientific discoveries and new directions in research take on a fresh importance.

Professor Strogatz is a teacher repeatedly honored by institutions and students alike. During his tenure at the Massachusetts Institute of Technology, he received the E. M. Baker Memorial Award for Excellence in Undergraduate Teaching, the university's only institute-wide teaching prize selected and awarded solely by students. In 2007, he received a lifetime achievement award for the communication of mathematics to the general public from the Joint Policy Board for Mathematics, which represents the four major American mathematical societies.

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.

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24 Lectures
  • 1
    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? x
  • 2
    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? x
  • 3
    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. x
  • 4
    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. x
  • 5
    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." x
  • 6
    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. x
  • 7
    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." x
  • 8
    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. x
  • 9
    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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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! x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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? x
  • 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. x
  • 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. x
  • 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 x

Lecture Titles

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Steven Strogatz
Ph.D. Steven Strogatz
Cornell University

Professor Steven Strogatz is the Jacob Gould Schurman Professor of Applied Mathematics and Professor of Theoretical and Applied Mechanics at Cornell University. He graduated summa cum laude from Princeton University with a B.A. in Mathematics and received his Ph.D. from Harvard University. Before joining Cornell University in 1994, Professor Strogatz was a faculty member at MIT. Professor Strogatz's books include Nonlinear Dynamics and Chaos-the most widely used textbook on chaos theory-and Sync: The Emerging Science of Spontaneous Order (chosen as a Best Book of 2003 by Discover magazine). Lauded for his exceptional teaching abilities, Professor Strogatz holds a Communications Award-a lifetime achievement award for the communication of mathematics to the general public-from the Joint Policy Board for Mathematics, which represents the four major American mathematical societies. He also received the Tau Beta Pi Excellence in Teaching Award from Cornell University's College of Engineering and the E. M. Baker Memorial Award for Excellence in Undergraduate Teaching from MIT.

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Reviews

Rated 4.6 out of 5 by 76 reviewers.
Rated 5 out of 5 Welcome to Chaos! Prof. Strogatz has opened up a really difficult area of study to mathematically-challenged people like me. Although mathematics is the basis of chaos theory, he has managed somehow to keep the math to a minimum and to get the concepts across, and with such clarity and enthusiasm! A really fine course. October 19, 2014
Rated 5 out of 5 by Chaos This wonderful course opened my eyes to fractals; it is a delightful glimpse into tomorrow-land! Thanks! July 12, 2014
Rated 5 out of 5 by Eager to teach Professor Strogatz is not only enthusiastic about the subject, but has a palpable eagerness to get YOU to understand what is so exciting about it. Throughout, I had the pleasant sensation of trusting the teacher not to lose me during complicated topics. The computer animations and physical demonstrations were especially helpful. The course covers a remarkable range of topics, giving a sense of how fertile this new approach is. Because of this, many lectures seem unfinished, with interesting topics only briefly introduced. This is not surprising for a 12 hour course. The course notes contain additional material not presented in the lectures, such as a glossary of terms and capsule biographies. June 1, 2014
Rated 5 out of 5 by Excellent Course for a Difficult Subject Several decades ago when I was working on my undergraduate work in Mathematics, I was continuously taught to keep the problem and solution within “well defined constraints”. While this works the majority of the time, it does not work all of the time because nature is not constrained by limits imposed by man. This had almost bothered me but nobody cause explain why except to say that the results are unpredictable. In this course, Professor Strogatz finally provided answers to those doubts that I have had for many years. These results are not unpredictable, they are non-deterministic (i.e., sometimes the answer is “a” and sometimes the answer is “b” but there is a probability for the “a” answer and for the “b” answer). This course material was presented in a very interesting and logical fashion. This course also shows that there is beauty in chaos as demonstrated on the lectures on fractals. You do not have to have any degrees or any background in Mathematics to understand and appreciate the concepts of this course. This course also provided me with excellent background material for two other excellent Great Courses; namely, the “Understanding the Universe: An Introduction to Astronomy, 2nd Edition” by Professor Filippenko and “Einstein's Relativity and the Quantum Revolution: Modern Physics for Non-Scientists, 2nd Edition” by Professor Wolfson. The information on non-deterministic results would very beneficial in understanding the concepts of Einstein’s theories of relativity and the general principles of quantum mechanics presented in these additional courses. This course is highly recommended. April 29, 2014
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