MLA 31500

Natural Sciences Elective

 

Order and Chaos in the Natural World

 

Spring Quarter 2008

 

CLASS NOTES

NINTH CLASS

May 24, 2008

 

I.     STEWART, CHAPTER 12: RETURN TO HYPERION

 

1.    The Newtonian two-body problem concerns the orbit of a star (Sun) and a single planet.  What is the relevance and role of the Newtonian two-body problem in connection with the investigations of the tumbling of Hyperion described by Stewart?  In connection with investigations of the orbits of asteroids in the Kirkwood gaps?  In connection with investigations of the stability of the solar system?

 

2.    In the model of the tumbling of Hyperion that underlies Figure 106 on page 234, friction is neglected.   Nevertheless, the role of friction is an important consideration in deciding that this is the appropriate model to investigate.  What is the role of friction in this case, and how does it constrain the choice of the model to be investigated?

 

3.    Is there a strange attractor in the chaotic region of the surface of section presented in Figure 106 on page 234?  If there were a strange attractor, how would it be identified?  Why might we conclude that there is no strange attractor here?

 

4.    The chaotic tumbling of Hyperion does not involve a strange attractor.  Why, nevertheless, would we conclude that the tumbling is chaotic?

 

II.   STEWART, CHAPTER 13: THE IMBALANCE OF NATURE

 

1.    The Oster model of population cycles includes cycles of periods three and six.  What might we expect in view of that information?

 

Recall the discussion of climate and weather by Lorenz.  The state of the system is described in terms of the values of quantities (e.g., temperature, pressure, wind speed and direction, relative humidity, etc.) that characterize the weather worldwide.  The picture is that the dynamics of the global weather system might be described in terms of an attractor – presumably a strange attractor.  If the atmosphere is approaching the attractor, then, although the weather is unpredictable in the long term, the attractor constrains the extremes of weather that might occur.  If the attractor is stable, then the long-term climate (defined by the attractor) is stable and predictable, notwithstanding that the weather is unpredictable.

 

2.    How does the picture described above for weather and climate translate into a characterization of population dynamics and an ecosystem as suggested in the section ÒThe Web of LifeÓ in Chapter 13?  What are the quantities that would characterize the state of an ecosystem?  What is the significance of the attractor?  In such terms, how might one characterize the transformation of a tropical rain forest into a desert?

 

3.    In the investigation of epidemics, how might one use the data illustrated in Figure 211 of Stewart in order to construct attractors like those illustrated in Figure 212?  And how might one construct a Poincare map from the result?

 

4.    As described by Stewart, is the kicked rotator a continuous dynamical system described by a differential equation or a system of differential equations, or is it a discrete dynamical system described by a difference equation or system of difference equations (i.e., an iterated map)?

 

 

LINKS:

 

Return to Course Page: mla315spring2008.htm

 

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Go to the home page of the Department of Astronomy and Astrophysics

of the University of Chicago:  http://astro.uchicago.edu/