Thu 10/30

MAIN POINTS
Chapter 5 examines higher-dimensional phase spaces due to linear systems. It begins with the simplest case— a two dimensional system. The text shows the form, and shows how it can be written in terms of a matrix A as x'=Ax. Fixed points also exist in 2D phase space... the simple harmonic oscillator in Ex5.1.1 oscillates in 2-space. Lines denote the orbits to form the Phase Portrait. In Ex5.1.2, the two equations are uncoupled, so may be solved separately. Stability is described in terms of 'attraction'— whether or not the system is attracted towards a fixed point for some initial condition. 5.2 looks at the more general case, which is similar to what we read about in the handout. It defines eigenvalues/vectors, etc. When the eigenvalues are complex, the FP is a center or spiral. Fig. 5.2.8 gives a great classification of the various stabilities based on trace and determinant. 5.3 examines love affairs between two Shakespearean characters. It discusses the character of the love based on parameters, and soves it as a 2x2 matrix system.

CHALLENGES
How would one illustrate a Liapunov-stable FP on a phase portrait?
What do systems with eigenvalue of multiplicity 2 look like? Drawing a phase portrait like that in Fig 5.2.2 looks really difficult... how would one do that?

REFLECTION
I'm taking linear algebra very late in my college career... I'm taking it concurrent with this course, so we're covering eigen* topics right now, conveniently. Two find an eigenvalue for an nxn matrix, we have to solve an nth degree polynomial, so how would we find the eigenvalue for a 4x4 or above? (Perhaps it's in the numerical analysis textbook...)