This section introduced linear systems. The solutions of a linear systems are trajectories moving in the (x,y) plane, which is called the phase plane in this context. The graph of the trajectories forms the phase portrait of a system. In these systems there are several types of fixed points. When the trajectories approach the fixed point it is labeled a stable node or a star. When there are infinite fixed points along a line it is called a line of fixed points. When the solutions are asymptotic away from the fixed point it is a saddle point, but there is a line that is stable called the stable manifold, and conversely there is an unstable manifold. You can analyze and classify linear systems by drawing its trajectories and manifolds.
Challenges
I wasn't sure how to graph the phase portrait of the system with complex eigenvalues(5.2.4 I believe). I tried to convert it into terms of sin and cos, and I think it is a clockwise spiral that grows away from the origin, but do I just draw an arbitrary spiral with these characteristics, or is there a more specific way?
Reflections
Love affairs. How cool is that application of linear systems? Determining how Romeo and Juliet will end up feeling about each other is definitely one of the most interesting things so far in the book. I like this chapter overall, too.
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