The Lorenz System and

Chaos in Nonlinear DEsApril 30, 2019

Math 333

p. 71 in Chaos: Making a New Science by James Gleick

Adding a dimension adds new possible layers of complexity in the phase space of a DE.

Today we’ll explore what can happen in 3+ dimensions!

Chaos

Chaos is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial

conditions.

Ex. The Butterfly effecthttps://en.wikipedia.org/wiki/

Butterfly#/media/File:Necyria_bellona_manco_NovaraExpZoologischeTheilLepido

pteraAtlasTaf36.jpg

Randomness is an indeterministic unpredictable process which may lead to a probability distribution

of outcome possibilities.

Randomness

https://en.wikipedia.org/wiki/Coin_flipping#/

media/File:Coin_Toss_(3635981

474).jpg

ex. Flipping a coin.

Philosophical Analogy: Randomness is to luck as chaos is to fate.

Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the atmospheric sciences, 20(2), 130-141.

dx

dt= �(y � x)

dy

dt= ⇢x� y � xz

dz

dt= ��z + xy

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The Lorenz System

• Very few nonlinear terms • Seems very simple

� = 10, � = 8/3, ⇢ = 28<latexit sha1_base64="ZMdEKtY7bDScV0znw+23YIaJfYc=">AAACDHicbVDLSgMxFM34rPVVdekmWAQXUmdawW4KRTcuK9gHdIaSSTNtaB5DkhHK0A9w46+4caGIWz/AnX9j2s5CWw8JnJx7Ljf3hDGj2rjut7Oyura+sZnbym/v7O7tFw4OW1omCpMmlkyqTog0YVSQpqGGkU6sCOIhI+1wdDOttx+I0lSKezOOScDRQNCIYmSs1CsUfU0HHNU89xz69oTEoFr1ojJ/qaGslavW5ZbcGeAy8TJSBBkavcKX35c44UQYzJDWXc+NTZAiZShmZJL3E01ihEdoQLqWCsSJDtLZMhN4apU+jKSyVxg4U393pIhrPeahdXJkhnqxNhX/q3UTE1WDlIo4MUTg+aAoYdBIOE0G9qki2LCxJQgrav8K8RAphI3NL29D8BZXXiatcsmrlMp3l8X6dRZHDhyDE3AGPHAF6uAWNEATYPAInsEreHOenBfn3fmYW1ecrOcI/IHz+QOHVJgb</latexit>

Famous parameter set:

(0, 0, 0), (±6p2,±6

p2, 27)

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Three equilibrium points:

At the origin:

J(x, y, z) =

2

4�10 10 028� z �1 �x

y x � 83

3

5

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At C+ = (6p2, 6

p2, 27)

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At C� = (�6p2,�6

p2, 27)

<latexit sha1_base64="M1UOf7s5cVUAgplFmWIcD/IIoOs=">AAACEXicbZC7SgNBFIZnvcZ4i1raDAYhggm7UYyNEE1jGcFcILuG2ckkGTJ7ceasGJa8go2vYmOhiK2dnW/j5FLExB8GPv5zDmfO74aCKzDNH2NhcWl5ZTWxllzf2NzaTu3sVlUQScoqNBCBrLtEMcF9VgEOgtVDyYjnClZze6VhvfbApOKBfwv9kDke6fi8zSkBbTVTGRvYI8SXgAelu+xFJntmq3sJcX5wPIX5wlEzlTZz5kh4HqwJpNFE5Wbq224FNPKYD1QQpRqWGYITEwmcCjZI2pFiIaE90mENjT7xmHLi0UUDfKidFm4HUj8f8MidnoiJp1Tfc3WnR6CrZmtD879aI4L2uRNzP4yA+XS8qB0JDAEexoNbXDIKoq+BUMn1XzHtEkko6BCTOgRr9uR5qOZz1kkuf3OaLl5N4kigfXSAMshCBVRE16iMKoiiJ/SC3tC78Wy8Gh/G57h1wZjM7KE/Mr5+AcKQm7A=</latexit>

p. 536 in Blanchard, Devaney and Hall

Strange Attractor

Strogatz: Numerical experiments suggest the strange attractor has fractal dimension 2.05.

From Lorenz’s original paper:

Sensitive Dependence on Initial ConditionsTwo ICs: (0,1,0) and (0,1.01,0)

from Sarah Iams, Harvard University

Sensitive Dependence on ICs Topological Mixing (Ergocity)

If this is interesting and you want to learn more:

1. Alligood, Kathleen T, Tim Sauer, and James A. Yorke. Chaos: An Introduction to Dynamical Systems. New York: Springer, 1997.

2. Blanchard, Paul, Robert L. Devaney, and Glen R. Hall. Differential Equations. Boston, MA: Brooks/Cole, Cengage Learning, 2012.

3. Gleick, James. Chaos: Making a New Science. New York, N.Y., U.S.A: Penguin, 1988. 4. Iams, Sarah. Lorenz Evolution.nb, (2019). 5. Lorenz, Edward N. "Deterministic nonperiodic flow." Journal of the atmospheric sciences 20.2 (1963):

130-141. 6. Strogatz, Steven. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry,

and Engineering. Boulder, Colo: Westview Press, 2014. 7. The Internet for some pictures :D 8. Weisstein, Eric W. "Lorenz Attractor." From MathWorld--A Wolfram Web Resource. http://

mathworld.wolfram.com/LorenzAttractor.html 9. Wikipedia: https://en.wikipedia.org/wiki/Randomness 10. Wikipedia: https://en.wikipedia.org/wiki/Edward_Norton_Lorenz

References