Computer Algebra Recipes

58 CHAPTER 2. PHASE-PLANE ANALYSIS
Problem 2-3: Vortex or focal point?
You are told that the following system has either a vortex or a focal point at
the origin. Analytically determine which it is and support your conclusion by
creating a suitable phase-plane portrait.
_x = y + x (x 2 + y 2 ); _y = ¡x + y (x 2 + y 2 )
2.1.2 The Mona Lisa of Nonlinear Science
Opinion is like a pendulum .... If it goes past the center of gravity on
one side, it must go a like distance on the other.
Arthur Schopenhauer, German philosopher (1788{1860)
In the world of art, one of the most famous portraits ever painted is that of
a woman with an enigmatic smile, referred to as the Mona Lisa. The artist
was the Italian Leonardo da Vinci (1452{1519), who was not only a painter,
but also a sculptor, architect, musician, and scientist. If Leonardo were alive
today, he would probably appreciate on esthetic, as well as scienti¯c, grounds
one of the most important phase-plane portraits of nonlinear science, that of
the simple plane pendulum. If we may make a puny pun, the plane pendulum
is not so plain as its name implies.
θ
L
mgsinθ
m
θ
mg
Figure 2.7: A simple plane pendulum.
What is a simple plane pendulum? It can be modeled as a small mass m
attached to the end of a very light rigid rod of length L that is allowed to
swing freely in a circular arc in the vertical plane as shown in Figure 2.7. Our