**MLP 31500**

**Natural Sciences Elective**

**Order and Chaos in the Natural World**

Spring Quarter 2014

CLASS NOTES

EIGHTH CLASS

May 20, 2014

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. LORENZ, CHAPTER 2: A JOURNEY INTO CHAOS

1. In this chapter, Lorenz introduces many of the principal concepts encountered in a general study of order and chaos. He does so with the aid of a particular set of dynamical models. It is illuminating that one such family of models can provide examples of so many elements of the subject. As you read the chapter, you might find it instructive to compile a list of the definitions, concepts, principles, tools, phenomenologies, etc. that Lorenz illustrates with the aid of these models.

2. Early in the chapter, Lorenz designs a computer model of a snowboard on a ski slope containing a regular array of moguls. What are the relevant physical laws underlying the model? What are the forces acting on the board? What are the variables in the phase space in which we describe the state of the system?

3. On page 38, Lorenz replaces the model of a ÒboardÓ with a model of a Òsled.Ó What is the difference between these two dynamical systems? Why does Lorenz turn to the model of the sled for his subsequent discussion?

4. On page 62, Lorenz returns to the model of the snowboard, but he makes the system frictionless, and he replaces the ski slope with a horizontal snowfield with the array of moguls. Why does he make these changes? What are the important consequences?

III. CONCEPTS INTRODUCED BY LORENZ (See Appendix 3 for a Glossary)

Dynamical system

State of a system; phase space

Continuous systems; flows and differential equations

Discrete systems; mappings and difference equations

Orbit

Deterministic System

Chaos

Limited chaos

Full chaos

Random process or random sequence

Sensitive dependence on initial conditions

Compact dynamical system

Periodic motion; almost periodic motion

Equilibrium

Stable equilibrium

Unstable equilibrium

Attractor

Limit point (fixed point)

Limit cycle

Strange attractor

Poincare maps (return maps); Poincare sections and surfaces of section

Cantor sets

Dissipative dynamical system

Basin of attraction; basin boundary

Non-dissipative dynamical system

Volume preserving system

Hamiltonian system

Poincare section for a Hamiltonian system

Sea of chaos

Periodic islands

Routes to chaos

Bifurcations

Period Doubling

**LINKS:**

Return to Course Page: mla315spring2013.html

Return to Peter Vandervoort's Home Page: pov.html

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

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